\section{Introduction}
An
\emph{antisymplectic involution} $\sigma$ on a complex K3-surface $X$ is a holomorphic involution acting nontrivially on the space $H^0(X,\Omega^2) = \mathbb C \cdot \omega $ of holomorphic 2-forms on $X$, i.e., $\sigma^* \omega = -\omega$.
Here we study K3-surfaces with antisymplectic involution from the point of view of symmetry and
consider actions of finite groups $G$ of symplectic transformations which are compatible with $\sigma$ in the sense that every $g \in G$ is a holomorphic automorphism of $X$ with $g^* \omega = \omega$ and $g \sigma = \sigma g$.
\subsection*{Main results}
Given a finite group $G$ we investigate if it can act in a compatible fashion on a K3-surface $X$ with antisymplectic involution
$\sigma$. If this is the case, then already the order of $G$ yields strong constraints on the geometry of $X$ and we obtain the following description.
\begin{theorem1}
Let $X$ be a K3-surface with a symplectic action of a finite group $G$ centralized by an
antisymplectic involution $\sigma$ and assume that
$\mathrm{Fix}_X(\sigma)\neq \emptyset$. If $|G|>96$, then $X/\sigma$ is a ($G$-minimal)
Del Pezzo surface and $\mathrm{Fix}_X(\sigma)$ is a smooth connected curve $B$ with $ g(B)\geq
3$.
\end{theorem1}
By a theorem due to Mukai \cite{mukai} finite groups of symplectic
transformations on K3-surfaces are characterized by the existence of a
certain embedding into a particular Mathieu group and are 
subgroups of the following eleven finite groups of maximal symplectic symmetry (cf.\ also \cite{kondo}).
\begin{table}[h]
\renewcommand{\arraystretch}{1.3}
\centering
\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c|c} 
   & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11  \\ \hline 
 group & $L_2(7)$& $A_6$& $S_5$& $M_{20}$& $F_{384}$& $A_{4,4}$& $T_{192}$& $H_{192}$ &$N_{72}$& $M_9$& $ T_{48}$ \\ \hline 
order & 168 & 360 & 120 & 960 & 384 & 288 & 192 & 192 & 72 &72& 48
\end{tabular}
\vspace{3mm}
\caption{Maximal finite groups of symplectic automorphisms on K3-surfaces}\label{TableMukai}
\end{table}
\vspace{-5mm}\\
\renewcommand{\arraystretch}{1}
This result naturally limits 
the possible choices of $G$ above
and has
led us to consider the classification problem for groups $G$ from this list of eleven groups:
as a refinement of Theorem \ref{roughclassi} we obtain the following complete classification.
\begin{theorem2}
Let $G$ be a group from Table \ref{TableMukai} acting on a K3-surface $X$ by symplectic transformations and let $\sigma $ be an antisymplectic involution on $X$ centralizing $G$ with $\mathrm{Fix}_X(\sigma) \neq \emptyset$. Then the pair $(X,G)$ is equivariantly isomorphic to a surface in Table \ref{Mukai times invol} below. In particular, for the groups $G$ numbered 4-8 on Mukai's list, there does not exist a K3-surface with an action of $G \times C_2$ satisfying the properties above.
\begin{table}[h]
\renewcommand{\arraystretch}{1.3}
\centering
\begin{tabular}{l|l|l}
  & $G$  & \textbf{K3-surface} $X$ \\ \hline 
1a & $L_2(7)$  & $\{x_1^3x_2+x_2^3x_3+x_3^3x_1+x_4^4 =0\} \subset \mathbb P_3$\\ \hline
1b & $L_2(7)$  & Double cover of $\mathbb P_2$ branched along 
		$\{x_1^5x_2+x_3^5x_1+x_2^5x_3-5x_1^2 x_2^2 x_3^2 =0\}$\\ \hline
2 & $A_6$  & Double cover of $\mathbb P_2$ branched along \\
& &  		$\{10 x_1^3x_2^3+ 9 x_1^5x_3 + 9 x_2^3x_3^3-45 x_1^2 x_2^2 x_3^2-135 x_1 x_2 x_3^4 + 27 			x_3^6  =0\}$\\ \hline
3a & $S_5$  & $\{\sum_{i=1}^5 x_i = \sum_{i=1}^6 x_1^2 = \sum_{i=1}^5 x_i^3=0\} \subset \mathbb P_5$\\ \hline
3b & $S_5$  & Minimal desingularization of the double cover of $\mathbb P_2$ branched along \\
 & &  $\{2x_1^4x_2x_3+2x_1x_2^4x_3+2x_1x_2x_3^4-2x_1^4x_2^2-2x_1^4x_3^2-2x_1^2x_2^4-2x_1^2x_3^4$\\
 & & $-2x_2^4x_3^2-2x_2^2x_3^4+2x_1^3x_2^3+2x_1^3x_3^3+2x_2^3x_3^3+x_1^3x_2^2x_3+x_1^3x_2x_3^2$\\
 & &  $+x_1^2x_2^3x_3+x_1^2x_2x_3^3+x_1x_2^3x_3^2 + x_1x_2^2x_3^3-6x_1^2x_2^2x_3^2=0\}$ \\ \hline
9 & $N_{72}$ & $\{ x_1^3+ x_2 ^3 + x_3^3 +x_4^3= x_1x_2 + x_3x_4+ x_5^2 = 0 \} \subset \mathbb P_4$ \\ \hline
10 & $M_9$ & Double cover of $\mathbb P_2$ branched along $\{x_1^6+x_2^6 +x_3^6 -10(x_1^3x_2^3 + x_2^3x_3^3 +x_3^3x_1^3) =0\}$\\ \hline
11a & $ T_{48}$  & Double cover of $\mathbb P_2$ branched along $\{x_1x_2(x_1^4-x_2^4)+ x_3^6 =0\}$\\ \hline
11b & $ T_{48}$  & Double cover of $\{  x_1x_2(x_1^4-x_2^4)+ x_3^3+x_4^2=0 \} \subset \mathbb P(1,1,2,3)$\\
&  		& branched along $\{x_3=0\} $
\end{tabular}
\vspace{2mm}
\caption{K3-surfaces with $G \times C_2$-symmetry}
\label{Mukai times invol}
\end{table}
\renewcommand{\arraystretch}{1}
\end{theorem2}
In addition to Examples 1a, 3a, 9, 10, and 11a, which have already been described by Mukai (Example 0.4 in \cite{mukai}) the table contains additional examples (1b, 2, 3b, and 11b) of K3-surfaces with maximal
symplectic symmetry. These arise as equivariant double covers of Del Pezzo surfaces.
\begin{remark}
Besides the above mentioned classification of finite groups of symplectic transformations on K3-surfaces by Mukai (cf.\ also Nikulin's classification in the Abelian case \cite{NikulinFinite} and Kond\=o's alternative approach using lattice theory \cite{kondo}) one of our starting points has been the study of K3-surfaces with $L_2(7)$-symmetry in \cite{OZ168}. Although our approach here is independent of the results by Keum, Oguiso, and Zhang (cf.\ e.g.\ \cite{OZ168}, \cite{KOZ1}, \cite{KOZ2}, and the summary \cite{ZhangSummary}), our understanding of finite group actions on K3-surfaces strongly gained from the numerous contributions to the subject by the authors above.
\end{remark}
We conclude this introduction with an outline of our general classification strategy.
\subsection*{Equivariant minimal model program}
The quotient of a K3-surface by an antisymplectic involution $\sigma$ with fixed points centralized by a finite group $G$ is a rational $G$-surface $Y$ to which we apply an equivariant version of the minimal model program respecting finite symmetry groups (cf.\ Example 2.18 in \cite{kollarmori} and Section 2.3  in \cite{Mori}). A detailed exposition using the language of Mori theory and including in particular equivariant analogues of the cone and contraction
theorems is given e.g.\ in \cite{kf}, Chapter 2 and \cite{FKH2}, Section 4. Here we briefly summarize the program in the following classification
result. 
\begin{proposition}\label{Gmmp}
Let $Y$ be a smooth projective surface with an action of a finite
group $G$.
Then there exists a sequence of $G$-equivariant
extremal contractions $Y \to Y_{(1)} \to \dots \to Y_{\mathrm{min}}$ such that
$Y_{\mathrm{min}}$ satisfies one of the following conditions:
\begin{enumerate}
\item
The canonical line bundle $\mathcal{K}_{Y_\mathrm{min}}$ is nef.
\item
$Y_\mathrm{min}$ is an $G$-equivariant conic bundle over a smooth curve,
i.e., there exists an $G$-equivariant morphism
$f: Y_{\mathrm{min}} \to C$ onto a smooth curve $C$ such that the general
fiber is a rational curve. If $f$ has singular fibers, these
consist of two (-1)-curves intersecting transversally.
\item
$\mathcal{K}^{-1}_{Y_{\mathrm{min}}}$ is ample, i.e., $Y_\mathrm{min}$ is a Del Pezzo surface.
\end{enumerate}
Each extremal contraction is the contraction of finitely many disjoint (-1)-curves forming a $G$-orbit.
\end{proposition}
The surface $Y_\mathrm{min}$ is referred to as a \emph{$G$-minimal model of $Y$},
and the map $Y \to Y_\mathrm{min}$ is called a \emph{Mori reduction}. 
 If $Y$ is a rational surface, then $Y_\mathrm{min}$ is either a Del Pezzo surface or an equivariant conic bundle over $\mathbb P_1$, i.e., the reduction leads to the well-known classification of $G$-minimal rational surfaces (\cite{maninminimal}, \cite{isk}).
\begin{remark}
Equivariant Mori reduction and the theory of $G$-minimal models have applications in various different contexts and can also be generalized to higher dimensions.
Initiated by Bayle and Beauville in \cite{Bayle}, the methods have been employed in the classification of subgroups of the Cremona group $\mathrm{Bir}(\mathbb P_2)$ for example by Beauville and Blanc (\cite{Beauville}, \cite{BeauBlancPrime}, \cite{PhDBlanc} \cite{Blanc1}), de Fernex \cite{fernex}, Dolgachev and Iskovskikh \cite{DolgIsk}, \cite{DolgIsk2}, and Zhang \cite{ZhangRational}. These references also provide details regarding the equivariant minimal model program summarized above.
\end{remark}
\subsection*{Del Pezzo surfaces}
In all cases under consideration in this article the surface $Y_\mathrm{min}$ is a Del Pezzo surface. 
Del Pezzo surfaces are classified according to their degree $1 \leq d \leq 9$, which is by definition the self-intersection number of the canonical divisor class. 
A Del Pezzo surface is either obtained from $\mathbb P_2$ by blowing up $9-d$ points in general position or is isomorphic to $\mathbb P_1 \times \mathbb P_1$.
The anticanonical map realizes a Del Pezzo surface of degree  $d \geq 3$ as a degree $d$ subvariety in $\mathbb P_d$, defines a double cover for $d=2$, and an elliptic fibration for $d=1$.

Our understanding of Del Pezzo surfaces as surfaces obtained by
blowing up points in $\mathbb P_ 2$ or as degree
$d$ subvarieties of $\mathbb P_d$ enables us to decide whether
a given finite group $G$ can occur as a subgroup of the automorphisms
group $\mathrm{Aut}(Z)$ of a certain Del Pezzo surface $Z$. Furthermore, we repeatedly consider the configuration of (-1)-curves on a Del Pezzo surfaces $Z$ and the induced action of $\mathrm{Aut}(Z)$ on it. 
In certain cases our analysis relies on Dolgachev's discussion of automorphism groups of Del Pezzo surfaces in \cite{dolgachev}, Chapter 10.

Using detailed knowledge of the equivariant reduction map $Y \to Y_\mathrm{min}$, the structure of the invariant set $\mathrm{Fix}_X(\sigma)$, 
and the equivariant geometry of Del Pezzo surfaces, we classify $Y_\mathrm{min}$, $Y$, and $\mathrm{Fix}_X(\sigma)$ and can describe $X$ as an equivariant double cover of a possibly blown-up Del Pezzo surface. 
\subsection*{Acknowledgements} 
The results presented in this note originate from the author's Ph.D.\ thesis at Ruhr-Universit\"at Bochum. The author would like to thank Alan Huckleberry for his advice and support and for numerous intensive, helpful discussions.

\section{Centralizers of antisymplectic involutions}\label{centralizer}
This section is dedicated to a rough classification (Theorem \ref{roughclassi}) of K3-surfaces with anti\-sym\-plec\-tic involutions centralized by large groups of symplectic transformations. For a group $H < \mathrm{Aut}(X)$ we denote by $H_\mathrm{symp} \lhd H$ the normal subgroup of symplectic automorphisms. The group $H / H_\mathrm{symp}$ acts effectively on $\Omega^2(X) = \mathbb C \cdot \omega_X$. In particular, if $H$ is finite, then $ H / H_\mathrm{symp}$ is cyclic.

We consider a K3-surface $X$ with an action of a finite group $G
\times C_2 < \mathrm{Aut}(X)$ and assume that $G < \mathrm{Aut}_\mathrm{symp}(X)$ whereas $C_2$ is generated by an
antisymplectic involution $\sigma$ centralizing $G$. Furthermore, we assume that $\mathrm{Fix}_X(\sigma) \neq \emptyset$. Let $\pi: X \to X/ \sigma = Y$ denote the quotient map, $R = \mathrm{Fix}_X(\sigma)$ and $B = \pi(R)$ the ramification and branch set of $\pi$. The quotient surface $Y$ is a smooth rational $G$-surface to which we apply an equivariant minimal model program. By Proposition \ref{Gmmp} a $G$-minimal model $Y_\mathrm{min}$ of $Y$ is either a Del Pezzo surface or an equivariant conic bundle over $\mathbb P_1$. In the later case, the following lemma shows that the possibilities for $G$ are limited by the classification of finite groups with an effective action on $\mathbb P_1$, i.e., cyclic and dihedral groups, $C_n$ and $D_{2n}$, and the exceptional tetrahedral, octahedral, and icosahedral group, $T_{12}$, $O_{24}$, $I_{60}$.
\begin{remark}\label{order of symp aut}
Several times we will use the fact that the order of a symplectic transformation on a K3-surfaces is at most eight. This follows implicitly from Table \ref{TableMukai}, but is originally due to an argument of Nikulin classifying finite
Abelian groups of symplectic transformation on K3-surfaces in \cite[\S5]{NikulinFinite}. Note that Nikulin' approach also shows that a symplectic automorphism of order 2,3,4,5,6,7,8 has 8,6,4,4,2,3,2 fixed points, respectively.
\end{remark}
\begin{lemma}\label{conicbundle}
If a $G$-minimal model $Y_\mathrm{min}$ of $Y$ is an equivariant conic bundle, then $|G| \leq 96$.
\end{lemma}
\begin{proof}
Let $\varphi: Y_\mathrm{min} \to \mathbb{P}_1$ be an equivariant conic bundle structure on $Y_\mathrm{min}$. We consider the induced action of $G$ on the base $\mathbb P_1$. If this action is effective, then $G$ is among the groups specified above. Since the maximal order of an element in $G$ is eight, 
it follows that the order of $G$ is bounded by 60.

If the action of $G$ on the base $\mathbb P_1$ is not effective, every element $n$ of the ineffectivity $N < G$ has two fixed points in the general fiber, which is isomorphic to $ \mathbb P_1$. This gives rise to a positive-dimensional $n$-fixed point set in $Y_\mathrm{min}$ and $Y$. A symplectic automorphism however has only isolated fixed points. It follows that the action of $n$ coincides with the action of $\sigma $ on $\pi^{-1}(\mathrm{Fix}_Y(N)) \subset X$ and the order of $n$ is two. Since $N$ acts effectively on the general fiber, it follows that $N$ is isomorphic to either $C_2$ or $C_2 \times C_2$. 

If $G/N$ is isomorphic to the icosahedral group $I_{60}= A_5$, then $G$ fits into the exact sequence $ 1 \to N \to G \to A_5 \to 1$ for $N=C_2$ or $C_2 \times C_2$. Let $\xi$ be an element of order five in $G$. Since neither $C_2$ nor $C_2 \times C_2$ has automorphisms of order five it follows that $\xi$ centralizes the normal subgroup $N$. In particular, there is a subgroup $C_2 \times C_5 \cong C_{10}$ in $G$ which contradicts the assumption that $G$ is a group of symplectic transformations and therefore its elements have order at most eight.

If $G/N$ is cyclic or dihedral, we again use the fact that the order of elements in $G$ is bounded by $8$ to conclude $|G/N| \leq 16$. It follows that the maximal possible order of $G/N$ is $|O_{24}|= 24$. Using $|N| \leq 4$ we obtain $|G| \leq 96$.
\end{proof}
For $| G | >96$ the lemma above allows us to restrict our classification to the case where any $G$-minimal model $Y_\mathrm{min}$ of $Y$ is a Del Pezzo surface.

\subsection{Branch curves and Mori fibers} \label{branch curves mori fibers}
We use the term \emph{curve} for irreducible 1-dimensional analytic subsets of surfaces.
For the remainder of this section we
fix an equivariant Mori reduction $M: Y \to Y_\mathrm{min}$.
A rational curve $E \subset Y$ is called a \emph{Mori fiber} if it is contracted in some step of the equivariant Mori reduction $Y \to Y_\mathrm{min}$. The set of all Mori fibers is denoted by $\mathcal E$. Its cardinality $|\mathcal E|$ is denoted by $m$.

A linearization argument shows that $\mathrm{Fix}_X(\sigma)$ is a disjoint union of smooth curves. More precisely, by a basic result originally due to Nikulin (\cite{NikulinFix}), the set $\mathrm{Fix}_X(\sigma)$ is either a union of two linearly equivalent elliptic curves or the union of one single curve of arbitrary genus with a (possibly empty) union of rational curves. In the following we let $n$ denote the total number of rational curves in $\mathrm{Fix}_X(\sigma)$. 
We choose a triangulation of $\mathrm{Fix}_X(\sigma)$ and extend it to a triangulation of the surface $X$. The topological Euler characteristic of the double cover is given by
\[
 e(X) = 24 = 2e(Y) - \sum_{C \subset \mathrm{Fix}_X(\sigma)} e(C)\geq 2e(Y) -2n  = 2(e(Y_\mathrm{min}) +m) -2n.
\]
This yields $m \leq n+12 - e(Y_\mathrm{min})$. Combining this with $e(Y_{min}) \geq 3$ shows that the total number $m$ of Mori fibers in $Y$ is bounded by 
\begin{equation}\label{moribound}
 m \leq n+ 12 - e(Y_\mathrm{min}) \leq n+9. 
\end{equation}
\begin{remark}\label{selfintcovering}
In the following, we repeatedly use the fact that for a finite proper surjective holomorphic map of complex manifolds (spaces) $\pi: X \to Y$ of degree $d$, the intersection number of pullback divisors fulfills $(\pi^*D_1 \cdot \pi^*D_2) =  d(D_1 \cdot D_2)$. In particular, if $\pi: X \to Y$ is a degree two map of surfaces, the image $\pi (C) \subset Y$ of a curve $C \subset X$ contained in the ramification locus $R$ of $\pi$ has self-intersection $(\pi(C))^2= 2 C^2$.
\end{remark}
\begin {lemma} \label {at most two}
Every Mori fiber $E \in \mathcal{E}$, $ E \not\subset B$ meets the branch locus $B$ in at
most two points. If $E$ and $B$ are tangent at $p$, then 
$E\cap B = \{p\}$ and $(E \cdot B)_p =2$.
\end {lemma}
\begin{proof} 
Let $E \in \mathcal{E}$, $E^2=k$ be a Mori fiber such that $E \not\subset B$ and $|E\cap B| \geq 2$ or $E \cdot B \geq 3$.
By the remark above, the divisor
$\pi^{-1}(E) = \pi^* E$ has self-inter\-section $2k$. Assume that $\pi^{-1}(E)$ is
reducible and let $\tilde E_1, \tilde E_2$ denote its irreducible components. These are rational
and therefore, by adjunction on the K3-surface $X$, have self-intersection number $-2$.  We write
$$
0 > 2k = (\pi^{-1}(E))^2 = \tilde E_1^2 + \tilde E_2^2 + 2 (\tilde E_1 \cdot \tilde E_2) = -4 + 2 (\tilde E_1 \cdot \tilde E_2).
$$
Since $\tilde E_1$ and $\tilde E_2$ intersect at points in the preimage of $E \cap B$, we obtain $\tilde E_1 \cdot \tilde E_2 \geq 2$, a contradiction. It follows that $\pi^{-1}(E)$ is irreducible. Consequently, $k=-1$ and $\pi^{-1}(E)$ is a smooth rational curve with precisely two $\sigma$-fixed points showing $|E\cap B| = 2$. It remains to show that the intersection is transversal.

To see this, let $N_{\tilde{E}}$ denote the normal bundle of $\tilde{E}$ in $X$. We
consider the induced action of $\sigma$ on $N_{\tilde{E}}$ by a bundle
automorphisms. We may equivariantly identify a tubular
neighbourhood of $\tilde E$ in $X$ with $N_{\tilde E}$ via a 
$C^{\infty}$-diffeomorphism. The $\sigma$-fixed point curves
intersecting $\tilde{E}$ map to curves of $\sigma$-fixed points in
$N_{\tilde{E}}$ intersecting the zero-section and vice versa. 
Let $D$ be a curve of $\sigma$-fixed points in $N_{\tilde{E}}$. If $D$ is
not a fiber of $N_{\tilde E}$, it follows that $\sigma$ stabilizes all 
fibers intersecting
$D$ and the induced action of $\sigma$ on the base must be trivial, a contradiction.
It follows that the $\sigma$-fixed point curves correspond to fibers of 
$N_{\tilde{E}}$, and $E$ and $B$ meet transversally.

By negation of the implication above, if $E$ and $B$ are tangent at $p$, then $|E \cap B |=1$ and $E \cdot B=2$.
\end{proof}
\begin{remark}\label{self-int of Mori-fibers}
Adjunction on $X$ and elementary considerations involving intersection numbers resembling those in the proof of the lemma above yield the following summary of the possible intersection geometry of a Mori fiber $E\in \mathcal E$ with 
the branch locus $B$ in relation to the self-intersection number of $E$.
\\
If $E \subset B $, then $E^2 = -4$. If $E \cap B = \emptyset$, then $E^2 = -2$ and $\pi^{-1}(E)$ is reducible.
If $E\cap B = \{p\}$, then $E^2 = -1$ and $\pi^{-1}(E)$ is reducible. If $E\cap B = \{p_1,p_2\}$, then $E^2 = -1$ and $\pi^{-1}(E)$ is irreducible.
More generally, the arguments involved apply to any (-1)-curve $E$ on $Y$ and therefore $E$ meets $B$ in either one or two points and $\pi^{-1}(E)$ is reducible or irreducible, respectively.
\end{remark}
\subsection{Rational branch curves}
In this subsection we find conditions on $G$, in particular conditions on the order of $G$, guaranteeing the absence of rational curves in $\mathrm{Fix}_X(\sigma)$. 

 It follows from adjunction that a curve with negative self-intersection on a Del Pezzo surface necessarily has self-intersection -1. So if $Y_\mathrm{min}$ is a Del Pezzo surface, all rational branch curves of $\pi$, which have self-intersection -4 by Remark \ref{selfintcovering}, need to be modified by the Mori reduction when passing to $Y_{\mathrm{min}}$ and therefore have nonempty intersection with the union of Mori fibers. We benefit from the following elementary observation regarding the behaviour of self-intersection numbers under monoidal transformations.
\begin{remark}\label{selfintblowdown} 
Let $\tilde X$ and $X$ be smooth projective surfaces and let $b: \tilde X \to X$ be the blow-down of a (-1)-curve $ E \subset \tilde X$. For a curve $B \subset \tilde X$, $B \neq E$, the self-intersection of its image in $X$ is given by $(b( B))^2 =  B^2 + ( E \cdot B)^2$.
\end{remark}
We denote by $\mathcal{C}$ the set of rational branch curves of
$\pi$. The total number $|\mathcal{C}|$ of these curves is denoted
by $n$. The union of all Mori fibers not contained in the branch locus $B$ is denoted by $\bigcup E_i$.
Let $\mathcal{C}_{\geq k}= \{ C \in \mathcal C \, | \, |C \cap  \bigcup E_i | \geq k\} $ be the set of those rational branch curves $C$ which meet $\bigcup E_i$ in at least $k$ distinct points and set $r_k = |\mathcal{C}_{\geq k}|$. 
We denote by
$\mathcal{E}_{\geq k}$ the set of Mori fibers $E \not\subset B$ which
intersect some $C \in \mathcal{C}_{\geq k}$ and define 
\[
P_k = \{ (p,E) \, |\, p \in C \cap E,\, E \in \mathcal{E}_{\geq k},\, C
\in \mathcal{C}_{\geq k}\} \subseteq Y \times \mathcal{E}_{\geq k}
\]
and the projection map $\mathrm{pr}_k: P_k \to \mathcal{E}_{\geq k}$ mapping
$(p,E)$ to $E$. This map is surjective by definition of
$\mathcal{E}_{\geq k}$ and its fibers consist of $\leq 2$ points by
Lemma \ref{at most two}. Using $|P_k| \geq k r_k$ we see
\begin{equation}\label{boundforE_k}
|\mathcal{E}_{\geq k}| \geq \frac{k}{2}r_k.
\end{equation} 
Let $N$ be the largest positive integer such that $\mathcal{C}_{\geq N} =
\mathcal{C}$, i.e., each rational ramification curve is intersected at least $N$
times by Mori fibers. 
A curve $C \in \mathcal{C}$ which is intersected precisely $N$ times by
Mori fibers is referred to as a \emph{minimizing curve}. In the
following, let $C$ be a minimizing curve and let $ H =
\mathrm{Stab}_G(C) < G$ be the stabilizer of $C$ in $G$.
The index of $H$ in $G$ is bounded by $n= r_N$.

The rational curves
in  $\mathrm{Fix}_X(\sigma)$ generate a sublattice of
$\mathrm{Pic}(X)$ of signature $(0,n)$ and it therefore follows immediately that $n
\leq 19$.
A sharper bound $n \leq 16$ for the number of disjoint (-2)-curves on a K3-surface has been
obtained by Nikulin \cite{NikulinKummer}.  In our setup an even sharper bound is due to Zhang \cite[Theorem 3]{ZhangInvolutions}, stating that
the total number of connected curves in the fixed point set of an antisymplectic
involution on a K3-surface is bounded by 10.
In the following, we use Zhang's bound for the number $n$ of rational curves in $\mathrm{Fix}_X(\sigma)$,
\begin{equation}\label{atmostten}
n \leq 10.
\end{equation}
Note, however, that all results can likewise be obtained by using the weakest bound $n \leq 19$. 
For $N \geq 4 $ Zhang's bound can be sharpened using the notion of
Mori fibers and minimizing curves.
Using inequalities \eqref{moribound} and \eqref{boundforE_k} we find
\begin{equation}\label{boundforn}
\frac{N}{2}n = \frac{N}{2}r_N \leq |\mathcal{E}_{\geq N}|\leq
|\mathcal{E}| \leq  n +12 - e(Y_\mathrm{min})  \leq n+9
\end{equation}
In the following we consider the stabilizer $H$ of a minimizing curve
$C$ and using the above inequalities for $n$, we obtain bounds for the order $|G|$
of $G$ guaranteeing the absence of rational curves in $\mathrm{Fix}_X(\sigma)$.
\begin{proposition}
Let $X$ be a K3-surface with an action of a finite group $G \times \langle \sigma
\rangle$ such that $ G < \mathrm{Aut}_\mathrm{symp}(X)$ and $\sigma$ is
an antisymplectic involution with fixed points. If $| G | > 108$,
then $\mathrm{Fix}_X(\sigma)$ does not contain rational curves.
\end{proposition}
\begin{proof}
 Let  $| G | > 108$ and assume on the contrary that $\mathrm{Fix}_X(\sigma)$ contains rational curves.
 We consider a minimizing curve $C \subset B$ and its stabilizer $\mathrm{Stab}_G(C) =:H$. Since a symplectic automorphism on $X$ does not admit a one-dimensional set of fixed points, it follows that the action of $H$ on $C$ is effective and $H$ is one the classical finite groups $C_n, D_{2n}, T_{12}, O_{24}, I_{60}$.
We recall the possible lengths of $H$-orbits in $C$: the length of an orbit of a dihedral group is at least two, the length of a $T_{12}$-orbit in $\mathbb P_1$ is at least four, the length of an $O_{24}$-orbit in $\mathbb P_1$ is at least six, and the length of an $I_{60}$-orbit in $\mathbb P_1$ is at least twelve.

 Let $Y_\mathrm{min}$ be a $G$-minimal model of $X/\sigma =Y$. Recall that by Lemma \ref{conicbundle} the surface $Y_\mathrm{min}$ is a Del Pezzo surface. Each rational branch curve is a (-4)-curve in $Y$. Since its image in $Y_\mathrm{min}$ has self-intersection $\geq -1$, it must intersect Mori fibers.
\begin{itemize}
 \item 
If $N=1$, i.e., the rational curve $C$ meets the union of Mori fibers in exactly one point $p$, then $p$ is a fixed point of the $H$-action on $C$. In particular, $H$ is a cyclic group $C_k$ and $k \leq 8 $ by Remark \ref{order of symp aut}. Since the index of $H$ in $G$ is bounded by $n \leq 10$, it follows that $|G | \leq 80$.
\item
If $N=2$, then $H$ is either a cyclic or a dihedral group. By Proposition 3.10 in \cite{mukai} the maximal order of a dihedral group of symplectic automorphisms on a K3-surface is 12. We first assume $H \cong D_{2m}$ and that the $G$-orbit $G.C$ of the rational branch curve $C$ has the maximal length $n=|G.C|=10$, i.e., $B = G . C$. 
Each curve in $G . C$ meets the union of Mori fibers in precisely two points forming an $D_{2m}$-orbit. If a Mori fiber $E_C$ meets the curve $C$ twice, then it follows from Lemma \ref{at most two} that $E$ meets no other curve in $B$. The contraction of $E$ transforms $C$ into a singular curve of self-intersection zero. The Del Pezzo surface $Y_\mathrm{min}$ does however not admit a curve of this type. It follows, that $E$ meets a Mori fiber $E'$ which is contracted in a later step of the Mori reduction and meets no other Mori fiber than $E'$.
The described configuration $G. E \cup G. E'$ requires a total number of at least 20 Mori fibers and therefore contradicts inequality \eqref{moribound}. If $C$ meets two distinct Mori fibers $E_1, E_2$, each of these two can meet at most one further curve in $B$. The contraction of $E_1$ and $E_2$ transforms $C$ into a (-2)-curve. As above, the existence of further Mori fibers meeting $E_i$ follows. Again, by invariance, the total number of Mori fibers exceeds 20, a contradiction. It follows that either $H$ is cyclic or $ |G.C|\leq 9$. Both cases imply $|G| \leq 108$. 
\item
If $N=3$, let $S=\{p_1,p_2,p_3\}$ be the points
of intersection of $C$ with the union $\bigcup E_i$ of Mori fibers. The set $S$ is
$H$-invariant. It follows that $H$ is either trivial or isomorphic to $C_2$, $C_3$
or $D_6$ and that $|G| \leq 60$
\item
If $N = 4$, it follows from inequality \eqref{boundforn} that $n \leq 9 $. Now $|H| \leq 12$ implies $|G| \leq 108$. 
\item
If $N=5$, the largest possible group acting on $C$ such
that there is an invariant subset of car\-di\-na\-li\-ty 5 is the dihedral
group $D_{10}$. Inequality \eqref{boundforn} implies $n \leq 6$, we conclude $|G| \leq 60$.
\item 
If $N=6$, then  $n \leq 4 $ and $|H| \leq 24$ implies $|G| \leq
96$. 
\item
If $N\geq 12$, then $n=1$ and $H =G$. The maximal order 60 is attained by
the icosahedral group. 
\item
If $6 < N <12$,
we combine $n \leq 4$ and $|H| \leq 24$ to obtain
$|G| \leq 96$.
\end{itemize}
The case by case discussion shows that the existence of a rational curve in $B$ implies $| G|\leq 108$ and  contradicts our assumption. The proposition follows.
\end{proof}
\begin{remark}
If the group $G$ under consideration is known not to contain certain subgroups (such as large dihedral groups or $T_{12}$, $O_{24}$ or $I_{60}$), then the condition $|G| >108$ in the proposition above can be improved and non-existence of rational ramification curves follows even for smaller groups $G$ (cf.\ Lemmata \ref{noratN72}, \ref{subgroupsM9}, and \ref{no rat T48}).
\end{remark}
\subsection{Elliptic branch curves}
The aim of this section is to determine conditions on the order of $G$ which allow us to exclude elliptic curves in $\mathrm{Fix}_X(\sigma)$. We prove:
\begin{proposition}\label{elliptic branch}
Let $X$ be a K3-surface with an action of a finite group $G \times \langle \sigma
\rangle$ such that $ G < \mathrm{Aut}_\mathrm{symp}(X)$ and $\sigma$ is
an antisymplectic involution with fixed points. If $| G | > 108$, 
then $\mathrm{Fix}_X(\sigma)$ contains neither rational nor elliptic ramification curves.
\end{proposition}
\begin{proof}
By the preceding proposition $\mathrm{Fix}_X(\sigma)$ does not contain rational curves. It follows from Ni\-ku\-lin's description of $\mathrm{Fix}_X(\sigma)$ in \cite{NikulinFix} that it is either a single curve of genus $g \geq 1$ or the disjoint union of two elliptic curves.

Let $T \subset B$ be an elliptic branch curve and let $H:= \mathrm{Stab}_G(T)$. If $H \neq G$, then $H$ has index two in $G$. The action of $H$ on $T$ is effective.
After conjugation the subgroup $H < \mathrm{Aut}(T)$ inherits the semidirect
product structure of $\mathrm{Aut}(T) = L \ltimes T$ for $L \in \{ C_2, C_4, C_6\}$. I.e.,
\[
H = (H \cap L) \ltimes  (H \cap T) .
\]
We refer to this decomposition as the \emph{normal form} of $H$.
By Lemma \ref{conicbundle} any $G$-minimal model of $Y$ is a Del Pezzo surface and therefore, by adjunction, does not admit elliptic curves with self-intersection zero. It follows that $T$ meets the union $\bigcup E_i$ of Mori fibers. Let $E$ be a Mori fiber meeting $T$. By Lemma \ref{at most two} their intersections fulfills $|T \cap  E |  \in \{1,2\}$. Since the total number of Mori fibers is bounded by 9 (cf.\ inequality \eqref{moribound}), the index of the stabilizer $\mathrm{Stab}_H(E)$ of $E$ in $H$ is bounded by 9.
If $T \cap E =\{p\}$, then $\mathrm{Stab}_H(E)$ is a cyclic group of order less than or equal to six and it follows that $|G | \leq 6\cdot 9 \cdot 2 = 108$.
If $T \cap E =\{p_1, p_2\}$, then $B \cap E = T \cap E$ and the stabilizer $\mathrm{Stab}_G(E)$ of $E$ in $G$ is contained in $H$. If both points $p_1,p_2$ are fixed by $\mathrm{Stab}_G(E)$, then $|\mathrm{Stab}_G(E)| \leq 6$. If $p_1,p_2$ form a $\mathrm{Stab}_G(E)$-orbit, then in the normal form $|\mathrm{Stab}_G(E) \cap T | =2$. It follows that $\mathrm{Stab}_G(E)$ is either $C_2$ or $D_4$. The index of $\mathrm{Stab}_G(E)$ in $G$ is bounded by 9 and $|G| \leq 54$.

In summary, the existence of an elliptic curve in $B$ implies $|G| \leq 108$.
\end{proof}
\subsection{Rough classification}
With the preparations of the previous subsections we may now turn to a classification result for K3-surfaces with antisymplectic involution centralized by a large group.
\begin{theorem}\label{roughclassi}
Let $X$ be a K3-surface with a symplectic action of a finite group $G$ centralized by an
antisymplectic involution $\sigma$ such that
$\mathrm{Fix}_X(\sigma)\neq \emptyset$. If $|G|>96$, then $X/\sigma$ is a $G$-minimal
Del Pezzo surface and $\mathrm{Fix}_X(\sigma)$ is a smooth connected curve $B$ with $ g(B)\geq
3$.
\end{theorem}
\begin{proof}
The group $G$ is a subgroup of one of the eleven maximal finite groups of symplectic transformations on Mukai's list \cite{mukai} (cf.\ Table \ref{TableMukai}).
None of these groups can have a subgroup $G$ with $96 < |G| < 120$.
In particular, the order of $G$ is at least 120.
We may therefore apply the results of the previous two sections and conclude that $\pi: X \to X/\sigma = Y$ is branched along a single smooth curve $B$ of general type. Its genus $g(B)$ must be greater than or equal to three by Hurwitz's formula.
It remains to show that $Y$ is $G$-minimal.

Assume on the contrary that $Y$ is not $G$-minimal. Then there exists a
Mori fiber $E \subset Y$ with $E^ 2= -1$.
By Remark \ref{self-int of Mori-fibers} it intersects the branch curve $B$ in one or two points. Let $\mathrm{Stab}_G(E)$ denote the stabilizer of $E$ in $G$. 

If $\pi^{-1}(E)$ is reducible its two irreducible components meet transversally in one point corresponding to $\{p\} =E \cap B$. The curve $E$ is tangent to $B$ at $p$ and we consider the linearization of the action of $\mathrm{Stab}_G(E)$ at $p$. If the action of $\mathrm{Stab}_G(E)$ on $E$ is not effective, the linearization of the ineffectivity $I < \mathrm{Stab}_G(E)$ yields a trivial action of $I$ on the tangent line of $B$ at $p$. It follows that the action of $I$ is trivial in a neighbourhood of $\pi^{-1}(p) \in R =\pi^{-1}(B)$. This is contrary to the assumption that $G$ acts symplectically on $X$.  
Consequently, the action of  $\mathrm{Stab}_G(E)$ on $E$ is effective and in particular, $\mathrm{Stab}_G(E)$ is a cyclic group. 

If $\pi^{-1}(E)$ is irreducible, then it is a smooth rational curve with an effective action of $\mathrm{Stab}_G(E)$. It follows that $\mathrm{Stab}_G(E)$ is either cyclic or dihedral. 

We conclude that the order of  $\mathrm{Stab}_G(E)$ is bounded by 12 and the index of $G_E$ in $G$ is strictly greater than nine. By inequality \eqref{moribound} the total number $m$ of Mori fibers however satisfies $m \leq 9$. This contradiction shows that $Y$ is $G$-minimal and, in particular, a Del Pezzo surface. 
\end{proof}
\begin{remark}\label{stab of minus one curve}
Let $X$ be a K3-surface with a symplectic action of $G$ centralized by an
antisymplectic involution $\sigma $ with $\mathrm{Fix}_X(\sigma) \neq \emptyset$ and let $E$ be a (-1)-curve on $Y = X/\sigma$. Then the argument above can be applied to see that the stabilizer of $E$ in $G$ is cyclic or dihedral and therefore has order at most 12.
\end{remark}
In the following section, the classification above is applied and extended to the case where $G$ is a maximal group of symplectic transformations on a K3-surface. 
\section{Maximal finite groups of symplectic transformations}\label{maximal}
In this section we consider K3-surfaces with a symplectic action of one of the eleven groups from Mukai's list (Table \ref{TableMukai}) centralized by an antisymplectic involution and prove the following classification result.
\begin{theorem}\label{thm mukai times invol}
Let $G$ be a group from Table \ref{TableMukai} acting on a K3-surface $X$ by symplectic transformations and $\sigma $ be an antisymplectic involution on $X$ centralizing $G$ with $\mathrm{Fix}_X(\sigma) \neq \emptyset$. Then the pair $(X,G)$ is equivariantly isomorphic to a surface in Table \ref{Mukai times invol}. In particular, for the groups $G$ numbered 4-8 on Mukai's list, there does not exist a K3-surface with an action of $G \times C_2$ satisfying the properties above. 
\end{theorem}
For the proof of this theorem we consider each group separately, show that any $G$-minimal model $Y_\mathrm{min}$ of the quotient surface $Y = X / \sigma$ is a Del Pezzo surface, and investigate which Del Pezzo surfaces admit an action of the group $G$.

It is then essential to study the branch locus $B$ of the covering $X \to Y$. As a first step, we exclude rational and elliptic curves in $B$ by studying their images in $Y_\mathrm{min}$ and their intersection with the union of Mori fibers. 
We then deduce that $B$ consists of a single curve of genus $\geq 2$ with an effective action of the group $G$. The possible genera of $B$ are restricted by the nature of the group $G$ and the Riemann-Hurwitz formula for the quotient of $B$ by an appropriate normal subgroup $N$ of $G$. The equations of $B$ or $X$ given in Table \ref{Mukai times invol} are derived using invariant theory.

Throughout the remainder of this chapter, the Euler characteristic formula
\begin{equation}\label{eulerchar}
 24 = e(X) = 2 e(Y_\mathrm{min}) + 2 m -2n +\underset{\text{if branch curve with }g\geq 2 \text{ exists}}{\underbrace{(2g-2)}}
\end{equation}
is exploited at various points.
Here $m$ denotes the total number of Mori contractions of the reduction $Y \to Y_ \mathrm{min}$, the total number of rational branch curves is denoted by $n$ and $g$ is the genus of a non-rational branch curve.
\subsection*{Equivariant equivalence}
Let us briefly formalize the notion of equivariant equivalence.
\begin{definition}\label{equivariantequivalence}
Let $(X_1, \sigma_1)$ and $(X_2, \sigma_2)$ be K3-surfaces with antisymplectic involution and let $G$ be a finite group acting on $X_1$ and $X_2$ by 
$\alpha_i: G \to \mathrm{Aut}_\mathrm{symp}(X_i)$,
such that $\alpha_i(g) \circ \sigma_i = \sigma_i \circ \alpha_i(g)$ for $i =1,2$ and all $g \in G$.
Then the surfaces $(X_1, \sigma_1)$ and $(X_2, \sigma_2)$ are considered \emph{equivariantly equivalent} or \emph{equivariantly isomorphic} if there exist a biholomorphic map $\varphi: X_1 \to X_2$ and a group automorphism $\psi \in \mathrm{Aut}(G)$ such that
\[
 \alpha_2(g) \varphi (x) = \varphi ( \alpha_1(\psi(g))x) \quad \text{and} \quad \sigma_2(\varphi (x)) = \varphi ( \sigma_1(x)).
\]
for all $x \in X_1$ and all $g \in G$.

Two (rational) surfaces $Y_1$ and $Y_2$
with actions 
$\alpha_i: G \to \mathrm{Aut}(Y_i)$ of a finite group $G$
are considered equivariantly equivalent if there exist a biholomorphic map $\varphi: Y_1 \to Y_2$ and a group automorphism $\psi \in \mathrm{Aut}(G)$ such that
$\alpha_2(g) \varphi (y) = \varphi ( \alpha_1(\psi(g))y)$
for all $y \in Y_1$ and all $g \in G$.
\end{definition}
\begin{remark}
If two K3-surfaces $(X_1, \sigma_1)$ and $(X_2, \sigma_2)$ are $G$-equivariantly equivalent, then the quotient surfaces $X_i / \sigma_i$ are equivariantly equivalent with respect to the induced action of $G$. 
\\
Conversely, let $Y$ be a rational surface with two actions of a finite group $G$ which are equivalent in the above sense and let $\varphi \in \mathrm{Aut}(Y)$ be the isomorphism identifying these two actions. We consider a smooth $G$-invariant curve $B$ linearly equivalent to $-2K_Y$ and the K3-surfaces $X_B$ and $X_{\varphi(B)}$ obtained as double covers branched along $B$ and $\varphi(B)$ equipped with their respective antisymplectic covering involution. 
If all elements of the group $G$ can be lifted to symplectic transformations on $X_B$ and $X_{\varphi(B)}$, then the central degree two extensions $E$ of $G$ acting on $X_B$, $X_{\varphi(B)}$, respectively, split as $E = E_\mathrm{symp} \times C_2$ with $E_\mathrm{symp} =G$.
In this case the group $G$ acts by symplectic transformations on 
$X_B$ and $X_{\varphi(B)}$ and
since for each $g \in G \subset \mathrm{Aut}(Y)$ 
there is only one choice of symplectic lifting $\tilde g \in E$
 the surfaces $X_B$ and $X_{\varphi(B)}$ are $G$-equivariantly equivalent in strong sense introduced above.
\end{remark}
\subsection{The group \texorpdfstring{$L_2(7)$}{L2(7)}}
Let $G \cong L_2(7) =\mathrm{PSL}(2, \mathbb F_7)=\mathrm{GL}_3(\mathbb F_2)$ be the finite simple group of order 168 acting on a K3-surface $X$. As $G$ is simple this action is effective and symplectic. Let $\sigma$ be an antisymplectic involution on $X$ centralizing $G$. Since $G$ has an element of order seven, which by Remark \ref{order of symp aut} has precisely three fixed points $p_1, p_2,p_3$ in $X$, and $\sigma$ acts on this set of three points, we know that $\mathrm{Fix}_X(\sigma) \neq \emptyset$.
I.e., the assumption $\mathrm{Fix}_X(\sigma) \neq \emptyset$ in Theorems \ref{roughclassi} and \ref{thm mukai times invol} is implicitly true for this particular group.

The classification result for the group $G = L_2(7)$ can be deduced from a description of K3-surfaces with $C_3 \ltimes C_7$-symmetry obtained in \cite{FKH1}, Theorem 2 (cf.\ also Chapter 5 in \cite{kf}, Theorem 5.5). Using the same arguments but restricting our consideration to the larger group $L_2(7)$ we present a simplified proof of the following proposition.
\begin{proposition}\label{classiL2(7)}
 Let $X$ be a K3-surface with a symplectic action of the group $L_2(7)$ centralized by an antisymplectic involution $\sigma$. Then $X$ is equivariantly isomorphic to either $C_\mathrm{Klein} = \{x_1^3x_2+x_2^3x_3+x_3^3x_1+x_4^4 =0\} \subset \mathbb P_3$ or the
double cover of $\mathbb P_2$ branched along $\mathrm{Hess}(C_\mathrm{Klein})$.
\end{proposition}
We begin with a classification of the quotient surface $Y = X / \sigma$.
\begin{lemma}
The quotient surface $Y$ is either $\mathbb P_2$ or a Del Pezzo surface of degree 2.
\end{lemma}
\begin{proof}
We may apply Theorem \ref{roughclassi} to conclude that $Y$ is a $G$-minimal Del Pezzo surface and denote 
by $d = \mathrm{deg}(Y) = K_Y^2$ its degree.

If $d=1$, then the anticanonical system $|-K_Y|$ is known to have precisely one base point. In particular, this point is fixed with respect to the full automorphism group.
Since $G$ does not admit a two-dimensional representation, this yields a contradiction.
As there is no injective homomorphism $G \hookrightarrow \mathrm{Aut}(\mathbb P_1 \times \mathbb
P_1) = (\mathrm{PSL}_2(\mathbb C) \times \mathrm{PSL}_2(\mathbb C)) \rtimes C_2$ and since the blow-up of $\mathbb P_2$ in one or two points admits an equivariant contraction map to either $\mathbb P_2$ or $\mathbb P_1 \times \mathbb P_1$ and is therefore never $G$-minimal, it follows that $d \neq 7,8$.  

For the remaining Del Pezzo surfaces we consider the action of $G$ on their configurations of (-1)-curves. 
Noting that the maximal
subgroups of $G$ are $C_3 \ltimes C_7$ and $S_4$ of index 8 and 7, respectively, we find that the $G$-orbit of a (-1)-curve $E
\subset Y$ consists of 7, 8, 14, 21, 24, 28 or more curves. Since the number of (-1)-curves on a Del Pezzo surface $Y$ of degree 3, 5, 6  equals  27, 10, 6, respectively, it follows that $d \not \in \{3,5,6\}$ .
If $d=4$, then the
union of (-1)-curves on $Y$ consists of two $G$-orbits of length 8. In
particular, $\mathrm{Stab}_G(E)\cong C_3 \ltimes C_7$ for any
(-1)-curve $E \subset Y$. 
Blowing down $E$ to a point $p \in Y'$ induces an action of $C_3 \ltimes C_7$ on $Y'$ fixing $p$. Since $C_3 \ltimes C_7$ does not admit a two-dimensional representation, it follows that the normal subgroup $C_7$ acts trivially on $Y'$ and therefore on $Y$. This is a contradiction and completes the proof of the lemma.
\end{proof}
It remains to classify the branch locus $B$ in both cases. 
\subsection*{Double covers of \texorpdfstring{$\mathbb P_2$}{P2}}
The action of $L_2(7)$ on $\mathbb P_2$ is necessarily given by a three-dimensional represention.
This follows from the fact that the group $L_2(7)$ does not admit nontrivial degree three central extensions. This can be derived from the cohomology group $H^2(L_2(7), \mathbb C^*) \cong C_2$ known as the Schur Multiplier. 
There are two isomorphism classes of three-dimensional representations and these differ by an outer automorphism. We may therefore consider one particular representation and check explicitly that in appropriately chosen coordinates the curve $B = \mathrm{Hess}(C_\mathrm{Klein}) = \{x_1^5x_2+x_3^5x_1+x_2^5x_3-5x_1^2 x_2^2 x_3^2 =0\}$ is $L_2(7)$-invariant. As the notation indicates this sextic is the Hessian curve associated to Klein's quartic curve $C_\mathrm{Klein}= \{ x_1^3x_2+x_2^3x_3+x_3^3x_1 =0\}$

In order to show that $B$ is the unique $G$-invariant sextic in $\mathbb P_2$ we consider the action of $G$ on $B$.
The maximal possible isotropy group is $C_7$ and each $G$-orbit in  $B$ consists of at least 21 elements. If there was another $G$-invariant smooth sextic curve $C \subset \mathbb P_2$, then the invariant set $ B \cap C$ would consist of at most 36 points. This is a contradiction.

Let $X$ be the K3-surface obtained as a double cover of $\mathbb P_2$ branched along $\mathrm{Hess}(C_\mathrm{Klein})$. 
It remains to check that $G$ lifts to a subgroup of $\mathrm{Aut}(X)$: On $X$ we find an action of a central degree two extension $E$ of $L_2(7)$. Since $L_2(7)$ is simple, it follows that $E_\mathrm{symp} \vartriangleleft E$ is mapped onto $L_2(7)$ and $E_\mathrm{symp} \cong L_2(7)$. In particular, the group $E$ splits as $E_\mathrm{symp} \times C_2$ where $C_2$ is generated by the antisymplectic covering involution. 
We have shown:
\begin{proposition}
 Let $X$ be a K3-surface with a symplectic action of the group $L_2(7)$ centralized by an antisymplectic involution $\sigma$. If $Y=X/\sigma \cong \mathbb P_2$ then $X$ is equivariantly isomorphic to the
double cover of $\mathbb P_2$ branched along $\mathrm{Hess}(C_\mathrm{Klein})$.
\end{proposition}
\subsection*{Double covers of Del Pezzo surfaces of degree two}
Here we assume that the degree of $Y$ equals two and consider the anticanonical map $Y \to \mathbb P_2$. It has degree two, is branched along a smooth curve of degree four, and equivariant with respect to $\mathrm{Aut}(Y)$. 
We obtain an action of $G$ on $\mathbb P_2$ stabilizing a smooth quartic. Choosing one particular 3-dimensional representation of $G$ one checks by direct computation that the smooth curve $C_\mathrm{Klein}= \{ x_1^3x_2+x_2^3x_3+x_3^3x_1 =0\}$ is invariant. It follows from classical invariant theory or from the intersection number argument applied above that $C_\mathrm{Klein}$ is the unique quartic curve in $\mathbb P_2$ with this property.
The curve $C_\mathrm{Klein}$ is a prominent example since its automorphism group $\mathrm{Aut}(C_\mathrm{Klein}) = G$ attains the maximal possible order $168 = 84(g(C_\mathrm{Klein}) -1)$ allowed by the Hurwitz formula.

Consider the branch curve $B \subset Y$ of the covering $\pi: X \to Y$ and the preimage $C \subset Y$ of $C_\mathrm{Klein}$. By construction, both curves belong to the linear system $|-2K_Y|$. If they do not coincide, they meet in $(-2K_Y)^2 = 8$ or less points. This contradicts the fact that the set $B \cap C$ must be $G$-invariant and therefore consists of at least 21 points in a $G$-orbit.

It follows that the surface $X$ is a cyclic degree four cover of $\mathbb P_2$ branched along $C_\mathrm{Klein}$ and can equivalently be described as the quartic hypersurface $\{x_1^3x_2+x_2^3x_3+x_3^3x_1+x_4^4 =0\} \subset \mathbb P_3$. On $X$ there is action of the group $\tilde G = G \times C_4$ where  $\tilde G_\mathrm{symp} = G$. We have shown:
\begin{proposition}
 Let $X$ be a K3-surface with a symplectic action of the group $L_2(7)$ centralized by an antisymplectic involution $\sigma$. If $Y=X/\sigma$ is a Del Pezzo surface of degree two then $X$ is equivariantly isomorphic to the
cyclic degree 4 cover of $\mathbb P_2$ branched along $C_\mathrm{Klein}$, i.e., Mukai's $L_2(7)$ example $\{x_1^3x_2+x_2^3x_3+x_3^3x_1+x_4^4 =0\} \subset \mathbb P_3$.
\end{proposition}
Combining this with the previous proposition yields the classification result summarized in Proposition \ref{classiL2(7)}. 
\begin{remark}
Proposition \ref{classiL2(7)} yields an alternative proof of the Main Theorem in \cite{OZ168} classifying K3-surfaces with actions of finite groups containing $G = L_2(7)$ as a subgroup of index four (cf.\ Corollary 1.3 in \cite{FKH1} and Theorem 6.1 and Lemma 6.2 in \cite{kf})
\end{remark}
\begin{remark}
In \cite{FKH1} K3-surfaces with a symplectic action of $C_3 \ltimes C_7$ centralized by an antisymplectic involution are described as double cover of $\mathbb P_2$ branched along sextic curves in a 1-dimensional family $\mathcal M$. Up to equivalence, there is a unique singular curve $C_\mathrm{sing}$ in this family. The K3-surface obtained as the minimal desingularization of the corresponding double cover of $\mathbb P_2$ is, by contruction, a double cover of the blow-up $b : Y \to \mathbb P_2$ at the seven singular points of $C_\mathrm{sing}$. The surface $Y$ is a Del Pezzo surface of degree two, its anticanonical map $\varphi: Y \to \mathbb P_2$ is branched along the proper transform $B$ of $C_\mathrm{sing}$ in $Y$ and its image $\varphi(B)$ is isomorphic to Klein's quartic curve. It follows that the surface $\{x_1^3x_2+x_2^3x_3+x_3^3x_1+x_4^4 =0\} \subset \mathbb P_3$ can also be characterized by the degeneration of the family $\mathcal M$ (cf.\ Theorem 1 in \cite{FKH1} and Theorem 5.4 in \cite{kf}). The blow-up map $ Y \to \mathbb P_2$ corresponds to the equivariant Mori reduction of $Y$ with respect to the subgroup $C_3 \ltimes C_7$ of $G$.
\end{remark}
\subsection{The group \texorpdfstring{$A_6$}{A6}}\label{A6Valentiner}
Let $G \cong A_6$ be the alternating group of degree 6 acting on a K3-surface $X$. Since $A_6$ is a simple group this action effective and symplectic. Let $\sigma$ be an antisymplectic involution on $X$ centralizing $G$ and assume that $\mathrm{Fix}_X(\sigma) \neq \emptyset$.  By Theorem \ref{roughclassi}, the K3-surface $X$ is a double cover of a Del Pezzo surface $Y$ with an effective action of $A_6$. 
\begin{lemma}
The Del Pezzo surface $Y$ is isomorphic to $\mathbb P_2$ with a uniquely determined action of $A_6$ given by the nontrivial central extension $V=3 .\ A_6$ of degree three known as \emph{Valentiner's group}. 
\end{lemma}
\begin{proof}
We go through the list of Del Pezzo surfaces according to their degree.

If $Y$ has degree one, then $| -K_Y|$ has precisely one base point which would have to be an $A_6$-fixed point. This is contrary to the fact that $A_6$ has no faithful two-dimensional representation. 

We recall that the stabilizer of a (-1)-curve $E$ in $Y$ is either cyclic or dihedral (Remark \ref{stab of minus one curve}). In particular, its order is at most 12 and therefore its index in $A_6$ is at least 30. This argument excludes Del Pezzo surfaces $Y$ of degree $\mathrm{deg}(Y) \in \{3,4,5,6,7\}$ since the number of (-1)-curves on $Y$ equals  27, 16, 10, 6, 3, respectively. Furthermore, the configuration of 56 exceptional curves on a Del Pezzo surface of degree 2 can neither be a single $A_6$-orbit nor the union of orbits and therefore  $\mathrm{deg}(Y) \neq 2$.

As was noted before the blow-up of $\mathbb P_2$ in one point is never $G$-minimal, hence
it remains to exclude $Y \cong \mathbb P_1 \times \mathbb P_1$. Assume there is an action of $A_6$ on $\mathbb P_1 \times \mathbb P_1$. Since $A_6$ has no subgroups of index two, it follows that $A_6 < \mathrm{PSL}_2(\mathbb C) \times  \mathrm{PSL}_2(\mathbb C)$ and both canonical projections are $A_6$-equivariant. This yields a contradiction since $A_6$ admits neither an effective action on $\mathbb P_1$ nor nontrivial normal subgroups of ineffectivity.

It follows that $Y \cong \mathbb P_2$. The action of $A_6$ on $\mathbb P_2$ is given by linear representation of a degree three central extension of $A_6$. Since $A_6$ has no faithful three-dimensional representation, this extension is nontrivial and isomorphic the unique nontrivial degree three extension $V=3.A_6$ known as Valentiner's group. Up to equivariant equivalence, there is a unique action of $A_6$ on $\mathbb P_2$. This follows from the classification of finite subgroup of $\mathrm{SL}_3(\mathbb C)$ (cf.\ \cite{blichfeldtbook}, \cite{blichfeldt}, and \cite{YauYu}) and can also be derived as follows: We need to show that any two actions induced by projective representations $\rho_i$ of $A_6$ are equivalent.
Restricting $\rho_i$ to the subgroup $A_5$ we obtain linear representation of $A_5$ and after a change of coordinates $\rho_1(A_5) = \rho_2(A_5) \subset \mathrm{SL}_3(\mathbb C)$. We fix a subgroup $A_4$ in $A_5$ and consider its normalizer $N$ in $A_6$. The groups $N$ and $A_5$ generate the full group $A_6$ and it suffices to prove that $\rho_1(N) = \rho_2(N)$. This is shown by considering an explicit three-dimensional representation of $A_4 < A_5$ and the normalizer $\mathcal N$ of $A_4$ inside $\mathrm{PSL}_3(\mathbb C)$. The group $A_4$ has index two in $\mathcal N$ and therefore $\mathcal N = \rho_1(N)= \rho_2(N)$.. 
\end{proof}
The covering $X \to Y$ is branched along an invariant curve $B$ of degree six. This curve is defined by an invariant polynomial $F_{A_6}$ of degree six, which is unique by Molien's formula. Its explicit equation is derived in \cite{crass}. In appropriately chosen coordinates,
\[
F_{A_6}(x_1,x_2,x_3) = 10 x_1^3x_2^3+ 9 x_1^5x_3 + 9 x_2^3x_3^3-45 x_1^2 x_2^2 x_3^2-135 x_1 x_2 x_3^4 + 27 x_3^6.
\]
The action of $A_6$ on $\mathbb P_2$ induces an action of a central degree two extension of $E$ on the double cover branched along $\{F_{A_6}=0\}$. 
By the same arguments as in the case $G = L_2(7)$ we may conclude that $E$ splits as $E_\mathrm{symp} \times C_2 = A_6 \times C_2$ where $C_2$ is generated by the antisymplectic covering involution. 
This proves the existence of a unique K3-surface with $A_6 \times C_2$-symmetry to which we refer as the \emph{Valentiner surface}. We have shown:
\begin{proposition}
Let $X$ be a K3-surface with a symplectic action of $A_6$ and let $\sigma$ be an antisymplectic involution on $X$ centralizing $A_6$ with $\mathrm{Fix}_X(\sigma) \neq \emptyset$. Then $X$ is equivariantly isomorphic to the Valentiner surface.
\end{proposition}
\subsection{The group \texorpdfstring{$S_5$}{S5}}
Let $X$ be a K3-surface with a symplectic action of $G = S_5$ and let $\sigma$ denote an antisymplectic involution centralizing $G$. We assume that $\mathrm{Fix}_X(\sigma) \neq \emptyset$. 
We may apply Theorem \ref{roughclassi} yielding that $X/ \sigma =Y$ is a $G$-minimal Del Pezzo surface and $\pi: X \to Y$ is branched along a smooth connected curve $B$ of genus $g(B) = 13-e(Y)$.
We show that only very few Del Pezzo surfaces admit an effective action of $S_5$ or a smooth $S_5$-invariant curve of appropriate genus.
\begin{lemma}
The degree $\mathrm{deg}(Y)=d$ of the Del Pezzo surface $Y$ is either three or five. 
\end{lemma}
\begin{proof}
We go through the list of $G$-minimal Del Pezzo surfaces and exclude all possible cases except $d \in \{3,5\}$.

Assume $Y \cong \mathbb P_2$, i.e., $G \hookrightarrow \mathrm{PSL}_3(\mathbb C)$, and let $\tilde G$ denote the preimage of $G$ in $\mathrm{SL}_3(\mathbb C)$. Since $A_5$ has no nontrivial central extension of degree three, it follows that the preimage of $A_5 < S_5$ in $\tilde G$ splits as $\tilde A_5 = A_5 \times C_3$. 
Let $g \in S_5$ be any transposition and pick $\tilde g$ in its preimage with $\tilde g^2 = \mathrm{id}$. Now $\tilde g$ and $ A_5$ generate a copy of $S_5$ in $\mathrm{SL}_3(\mathbb C)$. This is a contradiction since the irreducible representations of $S_5$ have dimensions $1,4,5$ or $6$ 

Next assume that $Y$ is isomorphic to $\mathbb P_1 \times \mathbb P_1$. We investigate the action of $S_5 = A_5 \rtimes C_2$ using the fact that $A_5$ is a simple group. Recalling $\mathrm{Aut}(Y) = (\mathrm{PSL}_2( \mathbb C) \times  \mathrm{PSL}_2(\mathbb C)) \rtimes C_2$ it follows that $A_5 < \mathrm{PSL}_2( \mathbb C) \times  \mathrm{PSL}_2( \mathbb C)$, i.e, the canonical projections onto the factors are $A_5$-equivariant.
If $A_5$ acts trivially on one of the factors, then it must act nontrivially on the second factor
and in particular the generator $\tau$ of the outer $C_2$ in $S_5$ stabilizes
the factors. It follows that $S_5$ 
acts effectively on the second factor. This is impossible since there is no effective action of $S_5$ on $\mathbb P_1$. We conclude that $A_5$ acts effectively on both factors and $\tau$ exchanges them.
We consider an element $\lambda$ of order five in $A_5$ and chose coordinates on $\mathbb P_1 \times \mathbb P_1$ such that $\lambda$ acts by
$([z_1:z_2],[w_1:w_2]) \mapsto ([\xi z_1:z_2],[\xi ^a w_1:w_2])$
for some $a \in \{1,2,3,4\}$ and $\xi^5 =1$. The automorphism $\lambda$ has four fixed points in $Y$.
Since it lifts to a symplectic automorphism on the K3-surface $X$ with four fixed points, all fixed points must lie on the branch curve.
The branch curve $B \subset Y$ is a smooth invariant curve linearly equivalent to $-2K_Y$ and is therefore given by an $S_5$-semi-invariant polynomial $f$ of bidegree $(4,4)$. Since $f$ must be invariant with respect to the commutator subgroup $S_5' = A_5$, it is a linear combination of $\lambda$-invariant monomials of bidegree $(4,4)$.
For each choice of $a$ one lists all $\lambda$-invariant monomials of bidegree $(4,4)$. In the case $a=1$ these are
$z_1 z_2^3 w_1^4,\,\, z_1^2 z_2^2 w_1^3 w_2 ,\,\, z_1^3 z_2 w_1^2 w_2^2,\,\,  z_1^4 w_1 w_2^3 , \,\,z_2^4 w_2^4$.
As $f$ must vanish at $p_1 \dots p_4$, one sees that it may not contain $z_2^4 w_2^4$. The remaining monomials have a common component $z_1 w_1$ such that $f$ factorizes and $B$ must be reducible, a contradiction.
The same argument can be carried out for each choice of $a$. It follows that the action of $S_5$ on  $\mathbb P_1 \times \mathbb P_1$ does not admit invariant irreducible curves of bidegree $(4,4)$. This eliminates the case $Y \cong \mathbb P_1 \times \mathbb P_1$.

Again using the fact that the largest subgroup of $S_5$ which can stabilize a (-1)-curve in $Y$ is the group $D_{12}$ of index 10,
it follows that the number of (-1)-curves in a $G$-orbit is at least 10
and Del Pezzo surfaces of degree four or six can be excluded by considering the action of $G$ on their set of (-1)-curves.

If $d =2$, then the anticanonical map defines an $\mathrm{Aut}(Y)$-equivariant double cover of $\mathbb P_2$. The induced action of $S_5$ on $\mathbb P_2$ would have to be effective and therefore we obtain a contradiction as in the case $Y \cong \mathbb P_2$.

If $d=1$ then the anticanonical system $|-K_Y|$ is known to have precisely one base point which has to be fixed point of the action of $S_5$. Since $S_5$ has no faithful two-dimensional representation, this is a contradiction. 
\end{proof}
\subsection*{Double covers of Del Pezzo surfaces of degree three}
The following example of a K3-surface $X$ with a symplectic action of $S_5$  can be found in Mukai's list \cite{mukai}. We identify an antisymplectic involution $\sigma$ on $X$ centralizing $S_5$ such that $X/\sigma$ is a Del Pezzo surface of degree three.
\begin{example}\label{MukaiS5}
Let $X$ be the K3-surface in $\mathbb P_5$ given by 
$\sum_{i=1}^5 x_i = \sum_{i=1}^6 x_1^2 = \sum_{i=1}^5 x_i^3=0$
 and let $S_5$ act on $\mathbb P_5$ by permuting the first five variables and by the character $\mathrm{sgn}$ on the last variable. This induces a symplectic action of $S_5$ action on $X$ (cf.\ \cite{mukai}).
The involution $\sigma : \mathbb P_5 \to \mathbb P_5$ given by $[x_1:x_2:x_3:x_4:x_5:x_6] \mapsto [x_1:x_2:x_3:x_4:x_5:-x_6]$ defines an antisymplectic involution on $X$ centralizing the action of $S_5$. 
The quotient $Y$ of $X$ by $\sigma$ is given by restricting then rational map $[x_1:x_2:x_3:x_4:x_5:x_6] \mapsto [x_1:x_2:x_3:x_4:x_5]$ to $X$. The surface $Y$ is given by 
$\{ \sum_{i=1}^5 y_i= \sum_{i=1}^5 y_i^3 =0\} \subset \mathbb P_4$
and is isomorphic to the Clebsch diagonal surface  
$\{z_1^2 z_2 + z_1 z_3^2 + z_3 z_4^2 + z_4 z_2^2 = 0\} \subset \mathbb P_3$ (cf.\ Theorem 10.3.10 in \cite{dolgachev}), a Del Pezzo surface of degree three. The branch set $B$ is given by $\{ \sum_{i=1}^5 y_1^2=0\} \cap Y \subset \mathbb P_4$.
By the following proposition, this example is the unique K3-surface with $S_5 \times \langle \sigma \rangle$-symmetry such that $X/\sigma$ is a Del Pezzo surface of degree three. 
\end{example}
\begin{proposition}\label{S5 on degree three}
Let $X$ be a K3-surface with a symplectic action of the group $S_5$ centralized by an antisymplectic involution $\sigma$. If $Y=X/\sigma$ is a Del Pezzo surface of degree three, then $X$ is equivariantly isomorphic to Mukai's $S_5$-example $\{\sum_{i=1}^5 x_i = \sum_{i=1}^6 x_1^2 = \sum_{i=1}^5 x_i^3=0\} \subset \mathbb P_5$. 
\end{proposition}
\begin{proof}
We consider the $\mathrm{Aut}(Y)$-equivariant embedding of the degree three Del Pezzo surface $Y$ into $\mathbb P_3$ given by the anticanonical map. Any automorphism of $Y$ is induced by an automorphism of the ambient projective space. 
It follows from the representation and invariant theory of the group $S_5$ that a Del Pezzo surface of degree three with an effective action of the group $S_5$ is equivariantly isomorphic the Clebsch cubic $\{z_1^2 z_2 + z_1 z_3^2 + z_3 z_4^2 + z_4 z_2^2 = 0\} \subset \mathbb P_3$ (cf.\ Theorems 10.3.9 and 10.3.10, Table 10.3 in \cite{dolgachev}). We show that the ramification curve $B \in |-2K_Y| $ is given by intersecting $Y$ with a quadric in $\mathbb P_3$. 

Applying the formula 
$h^0(Y, \mathcal{O}(-rK_Y))= 1+ \frac{1}{2}r(r+1) \cdot \mathrm{deg}(Y)$
(cf.\ e.g.\ Lemma 8.3.1 in \cite{dolgachev})
we obtain $h^0(Y,\mathcal{O}( -2K_Y))= 10$. This is also the dimension of the space of sections of $\mathcal O(2)$ in $\mathbb P_3$. It follows that the restriction map
$H^0(\mathbb P_3, \mathcal O (2)) \to H^0(Y, \mathcal O(-2K_Y))$
is surjective and $B = Y \cap Q$ for some quadric $Q = \{f=0\}$ in $\mathbb P_3$.

Since $B$ is an $S_5$-invariant curve in $Y$, it follows that for each $g \in S_5$ the intersection of $gQ = \{ f \circ g^{-1} =0\}$ and Y coincides with $B$. It follows that 
there exists a constant $c \in \mathbb C^*$ such that $(f \circ g^{-1}) - cf$ vanishes identically on $Y$. Since $Y$ is irreducible, this implies $f \circ g^{-1} = cf$. It follows that the polynomial $f$ is an $S_5$-semi-invariant and therefore invariant with respect to the commutator subgroup $A_5$.

We have previously noted that after a suitable linear change of coordinates the surface $Y$ is given by $\{ \sum_{i=1}^5 y_i= \sum_{i=1}^5 y_i^3 =0\} \subset \mathbb P_4$ where $S_5$ acts  by permutation. 
In these coordinates the semi-invariant polynomial $f$ is given by
$a \sum_{i=1}^5 y_i^2 + b(\sum_{i=1}^5 y_i)^2 =0$
for some $a,b \in \mathbb C$.
In particular,  $B = Y \cap \{f=0\} = Y \cap  \{\sum_{i=1}^5 y_i^2 =0\}$
and 
$X$ is Mukai's $S_5$-example discussed in Example \ref{MukaiS5}.
\end{proof}
\subsection*{Double covers of Del Pezzo surfaces of degree five}
A second class of candidates of K3-surfaces with $S_5 \times C_2$-symmetry is given by double covers of Del Pezzo surfaces of degree five. 
\begin{remark}
Any two Del Pezzo surfaces of degree five are isomorphic and the automorphisms group of a Del Pezzo surface $Y$ of degree five is $S_5$. The ten (-1)-curves on $Y$ form a graph known as the \emph{Petersen graph}. This graph has $S_5$-symmetry and every symmetry of the abstract graph is induced by a unique automorphism of the Del Pezzo surface. 
\end{remark}
\begin{proposition}\label{S5 on degree five}
Let $X$ be a K3-surface with a symplectic action of the group $S_5$ centralized by an antisymplectic involution $\sigma$. If $Y=X/\sigma$ is a Del Pezzo surface of degree five, then $X$ is equivariantly isomorphic to the minimal desingularization of the double cover of $\mathbb P_2$ branched along the sextic
\begin{align*}
 C_{S_5}=\{&2x_1^4x_2x_3+2x_1x_2^4x_3+2x_1x_2x_3^4-2x_1^4x_2^2-2x_1^4x_3^2-2x_1^2x_2^4-2x_1^2x_3^4\\
	   &-2x_2^4x_3^2-2x_2^2x_3^4+2x_1^3x_2^3+2x_1^3x_3^3+2x_2^3x_3^3+x_1^3x_2^2x_3+x_1^3x_2x_3^2\\
	   &+x_1^2x_2^3x_3+x_1^2x_2x_3^3+x_1x_2^3x_3^2+x_1x_2^2x_3^3-6x_1^2x_2^2x_3^2=0\}.
\end{align*}
\end{proposition}
\begin{proof}
Let $B \subset Y$ denote the branch locus of the covering $X \to Y$. The curve $B$ is invariant with respect to the full automorphism group of $Y$.
The Del Pezzo surface $Y$ is the blow-up $b: Y \to \mathbb P_2$ of four points $p_1,p_2,p_3,p_4 \in \mathbb P_2$ in general position. We may choose coordinates $[x_1:x_2:x_3]$ on $\mathbb P_2$ such that $p_1=[1:0:0],\ p_2=[0:1:0],\ p_3=[0:0:1],\ p_4=[1:1:1]$,
and consider the $S_4$-action on $\mathbb P_2$ permuting the points $\{p_i\}$. The isotropy at each point $p_i$ is isomorphic to $S_3$ and induces an effective $S_3$-action on $E_i =b^{-1}(p_i)$. 

Let $E$ be any (-1)-curve on $Y$. By adjunction $E\cdot B=2$. Since $Y$ contains precisely ten (-1)-curves forming an $S_5$-orbit, the group $H = \mathrm{Stab}_{S_5}(E)$ has order 12 and all stabilizer groups of (-1)-curves in $Y$ are conjugate.
It follows that the group $H$ contains $S_3$, which is acting effectively on $E$, and therefore $H$ is isomorphic to the dihedral group of order 12. The points of intersection $B\cap E$  form an $H$-invariant subset of $E$. Since $H$ has no fixed points in $E$ and precisely one orbit $H.p = \{p,q\}$ consisting of two elements, it follows that $B$ meets $E$ transversally in $p$ and $q$.
In particular, each curve $E_i$
meets $B$ in two points and the image $C = b(B)$ is a sextic curve with four nodes $\{p_1, \dots, p_4\}$. 
It is invariant with respect to the action of $S_4$ given by permutation on $p_1, \dots p_4$. For simplicity, we first only consider the action of $S_3$ permuting $p_1,p_2,p_3$ and conclude that $C$ is given by $\{f=\sum a_i f_i =0 \}$ for
\begin{align*}
f_1&=x_1^6+x_2^6 +x_3^6, &f_5=&x_1^3x_2^3+x_1^3x_3^3+x_2^3x_3^3, \\
f_2&=x_1^5x_2 + x_1^5x_3+ x_1x_2^5 +x_1x_3^5+x_2^5x_3+x_2x_3^6, &f_6=&x_1^3x_2^2x_3+x_1^3x_2x_3^2+x_1^2x_2^3x_3 \\ 
f_3&=x_1^4x_2x_3+x_1x_2^4x_3+x_1x_2x_3^4,  &&+x_1^2x_2x_3^3+x_1x_2^3x_3^2+x_1x_2^2x_3^3,\\
f_4&=x_1^4x_2^2+x_1^4x_3^2+x_1^2x_2^4+x_1^2x_3^4+x_2^4x_3^2+x_2^2x_3^4, 
&f_7=&x_1^2x_2^2x_3^2.
\end{align*}
The fact that $C$ passes through $p_i$ and is singular at $p_i$ yields $a_1=a_2=0$ and
$3a_3+6a_4+3a_5+6a_6+a_7=0$.
The two tangent lines of $C$ at the node $p_i$ correspond to the unique $\mathrm{Stab}(E_i)$-orbit of length two in $E_i$. We consider the point $p_3$ and the subgroup $S_3 < S_4$ stabilizing $p_3$. The action of $S_3$ on $E_3$ is given by the linearized $S_3$-action on the set of lines through $p_3$. One checks that in local affine coordinates $(x_1,x_2)$ the unique orbit of length two corresponds to the line pair $x_1^2 -x_1 x_2+x_2^2 =0$.
Dehomogenizing $f$ at $p_3$, i.e., setting $x_3=1$, we obtain the local equation $f_\mathrm{dehom}$ of $C$ at $p_3$. 
The polynomial $f_\mathrm{dehom}$ modulo terms of order three or higher must be a multiple of $x_1^2 -x_1 x_2+x_2^2 =0$. Therefore $a_3 = -a_4$.

Next we consider the intersection of $C$ with the line $L_{34} = \{x_1=x_2\}$ joining $p_3$ and $p_4$. We know that $f|_{L_{34}}$ vanishes of order two at $p_3$ and $p_4$ and at one or two further points on $L_{34}$. 
Let $\widetilde L_{34}$ denote the proper transform of $L_{34}$ inside the Del Pezzo surface $Y$. The curve $\widetilde L_{34}$ is a (-1)-curve, hence its stabilizer $\mathrm{Stab}_G(\widetilde L_{34})$ is isomorphic to $D_{12} = S_3 \times C_2$. The factor $C_2$ acts trivially on $\widetilde L_{34}$. Since the intersection of $\widetilde L_{34}$ with $B$ is $\mathrm{Stab}_G(\widetilde L_{34})$ invariant, it follows that $\widetilde L_{34} \cap B$ is the unique $S_3$-orbit a length two in $\widetilde L_{34}$.  
We wish to transfer our determination of the unique $S_3$-orbit of length two in $E_3$ above to the curve $\widetilde L_{34}$ using an automorphism of $Y$ mapping $E_3$ to $\widetilde L_{34}$. Consider the automorphism $\varphi$ of $Y$ induced by the birational map of $\mathbb P_2$ given by $[x_1:x_2:x_3] \mapsto [x_1(x_3-x_2):x_3(x_1-x_2):x_1x_3]$
(cf.\ Theorem 10.2.2 in \cite{dolgachev}) and let $\psi$ be the automorphism of $Y$ induced by the permutation of the points $p_2$ and $p_3$ in $\mathbb P_2$. Then $\psi \circ \varphi$ is an automorphism of $Y$ mapping $E_3$ to $\widetilde L_{34}$. If $[X_1:X_2]$ denote homogeneous coordinates on $E_3$ induced by the affine coordinates $(x_1,x_2)$ in a neighbourhood of $p_3$, then a point $[X_1:X_2] \in E_3$ is mapped to the point corresponding to $[X_1:X_1: X_1-X_2] \in L_{34} \subset \mathbb P_2$. It was derived above that the unique $S_3$-orbit of length two in $E_3$ is given by $X_1^2-X_1X_2 +X_2^2$ and it follows that the unique $S_3$-orbit of length two in $\widetilde L_{34}$ corresponds to the points $[x_1:x_1:x_3] \in \mathbb P_2$ fulfilling $x_1^2 -x_1x_3 +x_3^2 =0$. 
Therefore, $f|_{L_{34}}$ is a multiple of polynomial given by $x_1^2(x_1-x_3)^2(x_1^2 -x_1x_3 +x_3^2)$.
Comparing coefficients with $f(x_1:x_1:x_3)$ yields
\begin{align*}
2a_3+2a_6 &= 2a_5 +2a_6, & 2a_4 + a_5&= 2a_4+a_3,\\
8a_4+4a_5 &= 2a_4+2a_6 +a_7, &-6a_4 -3a_5 &= 2a_5 +2a_6.
\end{align*}
We conclude $a_3=a_5= -a_4=2$, $a_6=1$, and $a_7=-6$. So if $X$ as in the proposition exists, it is the double cover of $Y$ branched along the proper transform of $\{f = 2f_3 -2f_4+2f_5+f_6-6f_7  = 0\}$ in $Y$.

In order to prove existence, let $X$ be the minimal desingularization of the double cover of $\mathbb P_2$ branched along $\{f=0\}$. Then $X$ is the double cover of the Del Pezzo surface $Y$ of degree five branched along the proper transform $D$ of $\{f=0\}$ in $Y$.
Since all automorphisms of $Y$ are induced by explicit biholomorphic or birational transformation of $\mathbb P_2$ 
one can check by direct computations that $D$ is in fact invariant with respect to the action of $\mathrm{Aut}(Y) = S_5$.

On $X$ there is an action of a central extension $E$ of $S_5$, $\{\mathrm{id}\} \to C_2 \to E \to S_5 \to \{\mathrm{id}\}$. Let $E_\mathrm{symp}$ be the subgroup of symplectic automorphisms in $E$. Since $E$ contains the antisymplectic covering involution $E_\mathrm{symp} \neq E$. The image $N$ of $E_\mathrm{symp}$ in $S_5$ is normal and therefore either $N \cong S_5$ or $N \cong A_5$.
If $N \cong A_5$ and $| E_\mathrm{symp}| =60$, then  $E_\mathrm{symp} \cong A_5$. Lifting any transposition from $S_5$ to an element $g$ of order two in $E$, the group generated by $g$ and $E_\mathrm{symp}$ inside $E$ is isomorphic to $S_5$. It follows that $E$ splits as $S_5 \times C_2$ and $E / E_\mathrm{symp} \cong C_2 \times C_2$. Since this is not cyclic, we obtain a contradiction.  
If $N \cong A_5$ and $| E_\mathrm{symp}| =120$, then $ E = E_\mathrm{symp} \times C_2$, where the outer $C_2$ is generated by the antisymplectic covering involution $\sigma$, and $E / C_2 = S_5$ implies that $E_\mathrm{symp} \cong S_5$. This is contradictory to the assumption $N \cong A_5$.
In the last remaining case $N \cong S_5$. Since $E_\mathrm{symp} \neq E$, also $E_\mathrm{symp} \cong S_5$ and $E$ splits as $E_\mathrm{symp} \times C_2$. 
It follows that the action of $S_5$ on $Y$ induces an symplectic action of $S_5$ on the double cover $X$ centralized by the antisymplectic covering involution. This completes the proof of the proposition.
\end{proof}
We summarize Propositions \ref{S5 on degree three} and \ref{S5 on degree five} as follows.
\begin{proposition}
Let $X$ be a K3-surface with a symplectic action of the group $S_5$ centralized by an antisymplectic involution $\sigma$ with $\mathrm{Fix}_X(\sigma) \neq \emptyset$. Then $X$ is equivariantly isomorphic to either Mukai's $S_5$-example or the minimal desingularization of the double cover of $\mathbb P_2$ branched along the sextic $C_{S_5}$.
\end{proposition}
\subsection{The group \texorpdfstring{$M_{20} = C_2^4 \rtimes A_5$}{M20}}
We assume that a K3-surface $X$ with a symplectic action of $M_{20}$ centralized by an antisymplectic involution $\sigma$ and $\mathrm{Fix}_X(\sigma) \neq \emptyset$ exists. Applying Theorem \ref{roughclassi} we see that $X \to Y$ is branched along a single $M_{20}$-invariant smooth curve $B$ with $g(B) \geq 3$ on the Del Pezzo surface $Y$. By Hurwitz's formula, $| \mathrm{Aut} (B) | \leq 84(g(B)-1)$,
the genus of $B$ must be at least twelve. As $B$ is linearly equivalent to $-2K_Y$, the adjunction formula 
$2g(B)-2 = (K_Y,B) + B^2 = 2K_Y^2 $
implies $\mathrm{deg}(Y) = K_Y^2 \geq 11$. Since the degree of a Del Pezzo surface is at most nine this is a contradiction yielding the following non-existence result. 
\begin{proposition}
There does not exist a K3-surface with a symplectic action of $M_{20}$ centralized by an antisymplectic involution $\sigma$ with $\mathrm{Fix}_X(\sigma) \neq \emptyset$.
\end{proposition}
\subsection{The group \texorpdfstring{$F_{384} = C_2^4 \rtimes S_4$}{F384}}
Before we prove non-existence of K3-surfaces with $F_{384} \times C_2$-sym\-me\-try, we note the following useful fact about $S_4$-actions on Riemann surfaces.
\begin{remark}\label{S4 not on g=1,2}
The group $S_4$ does not admit an effective action on a Riemann surface of genus one or two.
The first case can be excluded by using the explicit shape of the automorphism group of a Riemann surface $T$ of genus one, $\mathrm{Aut}(T)= L \ltimes T$ for $L \in \{C_2, C_4, C_6\}$. 
The second case is excluded by considering the quotient by the hyperelliptic involution branched at six points.
Since $S_4$ has no normal subgroup of order two, the induced action of $S_4$ on the quotient $\mathbb P_1$ is effective and therefore has precisely one orbit consisting of six points. 
The isotropy group at the corresponding points in the hyperelliptic curve would be isomorphic to $C_4 \times C_2$, which is contradiction.
\end{remark}
\begin{proposition}
There does not exists a K3-surface with a symplectic action of $F_{384}$ centralized by an antisymplectic involution $\sigma$ with $\mathrm{Fix}_X(\sigma) \neq \emptyset$.
\end{proposition}
\begin{proof}
As above, assume that a K3-surface $X$ with these properties exists and apply Theorem \ref{roughclassi} to see that $X \to Y$ is branched along a single $F_{384}$-invariant smooth curve $B$ on the Del Pezzo surface $Y$. By Hurwitz's formula the genus of $B$ is at least 6.
It now follows from adjunction, $K_Y^2 = g(B)-1$, that the degree of the Del Pezzo surface $Y$ is at least five.
We consider the action of $F_{384}$ on the configuration of (-1)-curves on $Y$ and recall that the order of a stabilizer of a (-1)-curve in $Y$ is at most twelve (cf.\ Remark \ref{stab of minus one curve}) and therefore has index greater than or equal to $32$ in $G$. It follows that $Y$ is either $\mathbb P_1 \times \mathbb P_1$ or $\mathbb P_2$. In the first case, the canonical projections of $\mathbb P_1 \times \mathbb P_1$ are equivariant with respect to a subgroup of index two in $F_{384}$ and thereby contradict Lemma \ref{conicbundle}. Consequently, $Y \cong \mathbb P_2$
and $g(B) =10$.

We use the realization of $F_{384}$ as a semi-direct product $C_4^2 \rtimes S_4$ (cf.\ \cite{mukai}) and consider the quotient $Q$ of the branch curve $B$ by the normal subgroup $N = C_4^2$. On $Q$ there is the induced action of $S_4$. It follows from the remark above that $Q$ is either rational or $g(Q) >2$. In the second case the Riemann-Hurwitz formula applied to the covering $ B \to Q$,
\[
-18 = e(B) = 16 e(Q) - \text{branch point contributions} \leq -64,
\]
yields a contradiction.
It follows that $Q$ is a rational curve and that the branch point contribution of the covering $B \to Q$ is 50. Since isotropy groups at points in $B$ must be cyclic, the only possible isotropy subgroups of $N = C_4^2$ at points in $B$ are $C_2$ and $C_4$ and have index four or eight. The full branch point contribution must therefore be a multiple of four. This contradiction yields the non-existence claimed. 
\end{proof}
\subsection{The group \texorpdfstring{$A_{4,4} = C_2^4 \rtimes A_{3,3}$}{A4,4}}
By $S_{p,q}$ for $p+q =n$ we denote a subgroup $S_p \times S_q$ of $S_n$ preserving a partition of the set $\{1,\dots, n\}$ into two subsets of cardinality $p$ and $q$. The intersection of $S_{p,q}$ with $A_n$ is denoted by $A_{p,q}$.
\begin{proposition}
There does not exists a K3-surface with a symplectic action of $A_{4,4}$ centralized by an antisymplectic involution $\sigma$ with $\mathrm{Fix}_X(\sigma) \neq \emptyset$.
\end{proposition}
\begin{proof}
We again assume that a K3-surface with these properties exists and applying Theorem \ref{roughclassi} we see that $X \to Y$ is branched along a single $A_{4,4}$-invariant smooth curve $B$ on the Del Pezzo surface $Y$. The group $A_{4,4}$ is a semi-direct product $C_2^4 \rtimes A_{3,3}$ (see e.g.\ \cite{mukai}). We consider the quotient $Q$ of $B$ by the normal subgroup $N \cong C_2^4$. On $Q$ there is an action of $A_{3,3}$.
Since $A_{3,3}$ contains the subgroup $C_3 \times C_3$,
it follows that $Q$ not rational. In particular, $e(Q) \leq 0$. We apply the Riemann-Hurwitz formula to the covering $B \to Q$,
\[
2-2g(B) = e(B) = 16 e(Q) - \text{branch point contributions} \leq  - \text{branch point contributions}.
\]
As above, isotropy groups must be cyclic and the maximal possible isotropy group of the $C_2^4$-action on $B$ is $C_2$, which has index eight in $C_2^4$. Consequently, the branch point contribution at each branch point is eight.
The action of $C_3 \times C_3 < A_{3,3}$ on $Q$ has orbits of length greater than or equal to three. Therefore, the total branch point contribution must be greater than or equal to $24$. In particular, $g(B) = \mathrm{deg}(Y) +1 \geq 13$ contrary to $\mathrm{deg}(Y) \leq 9$.
\end{proof}
\subsection{The groups \texorpdfstring{$T_{192} = (Q_8 * Q_8) \rtimes S_3$}{T192} and \texorpdfstring{$H_{192} = C_2^4  \rtimes D_{12}$}{H192}}
By $Q_8$ we denote the quaternion group $\{\pm 1, \pm I, \pm J, \pm K\}$ where $I^2= J^2= K^2 = IJK = -1$. 
The central product $ Q_8 * Q_8 $ is defined as the quotient of $Q_8  \times Q_8$ by the central involution $(-1, -1)$.
Most importantly, we benefit from the observation that
both groups $T_{192}$ and $H_{192}$ are semi-direct products $C_2^3 \rtimes S_4$ (cf.\ \cite {mukai}).

We prove the following non-existence result. 
\begin{proposition}\label{192}
For $G =T_{192}$ or $G = H_{192}$ there does not exists a K3-surface with a symplectic action of $G$ centralized by an antisymplectic involution $\sigma$ with $\mathrm{Fix}_X(\sigma) \neq \emptyset$.
\end{proposition}
Assume that a K3-surface as in the proposition exists. Applying Theorem \ref{roughclassi} we see that $X \to Y$ is branched along a single $G$-invariant smooth curve $C$ on the Del Pezzo surface $Y$. The genus of $C$ is at least four by Hurwitz's formula and therefore $d = \mathrm{deg}(Y) \geq 3$.
\subsection*{The cases \texorpdfstring{$3 \leq  d \leq 7$}{d=3,4,5,6,7}}
We consider the action of $G$ on the Del Pezzo surface $Y$ of degree $\geq 3$, in particular the induced action on its configuration of (-1)-curves. By Remark \ref{stab of minus one curve} 
the stabilizer of a (-1)-curve in $Y$ has index $\geq 16$ in $G$ and we may immediately exclude the cases $\mathrm{deg}(Y) = 3,5,6,7$.
The automorphism group of a Del Pezzo surface of degree four is $C_2^4 \rtimes \Gamma$ for $\Gamma \in \{C_2, C_4, S_3, D_{10}\}$ (cf.\ Section 10.2.2 in \cite{dolgachev}). In particular, the maximal possible order is 160 and therefore $\mathrm{deg}(Y) \neq 4$.
It remains to consider the cases $Y \cong \mathbb P_2$ and $Y \cong \mathbb P_1 \times \mathbb P_1$
\subsection*{The case \texorpdfstring{$Y \cong \mathbb P_2$}{Y=P2}}
If $Y \cong \mathbb P_2$ than $g(C) =10$. 
We consider the quotient $Q$ of $C$ by the normal subgroup $N = C_2^3$ and obtain
\[
 -18 = e(C) = 8\cdot e(Q) - \text{branch point contributions}.
\]
By Lemma \ref{S4 not on g=1,2} 
the quotient $Q$ is either rational or $g(Q) >2$, i.e., $e(Q) \leq -4$. Since the second case is impossible it follows that $Q$ is a rational curve and the
branch point contribution must be $34$. The maximal possible isotropy subgroup of $N = C_2^3$ at a point in $C$ is $C_2$ and the full branch point contribution must be divisible by four. This is a contradiction.
\subsection*{The case \texorpdfstring{$Y \cong \mathbb P_1 \times \mathbb P_1$}{Y = P1 times P1}}
Assume that $Y \cong \mathbb P_1 \times \mathbb P_1$. The canonical projection $\pi_{1,2}: Y \to \mathbb P_1$ is equivariant with respect to a subgroup $\tilde G$ of $G$ of index at most two. It follows that $\tilde G$ fits into the exact sequences
$ \{\mathrm{id}\} \to I_i \to \tilde G \overset{(\pi_i)_*}{\to} H_i\to  \{\mathrm{id}\}$, 
where $I_i \cong C_2 \times C_2$ is the ineffectivity of the induced $H$-action on the base $\pi_i(Y)$ and $H_i \cong S_4$ (cf.\ proof of Lemma \ref{conicbundle}).
We consider the realization of $G$ as a semi-direct product $C_2^3 \rtimes S_4 = N \rtimes S_4$ and denote by $A = \tilde G \cap N$ the intersection of the normal subgroup $N$ with the index two normal subgroup $\tilde G$.
\begin{lemma}
The group $A$ is isomorphic to $C_2 \times C_2$.
\end{lemma}
\begin{proof}
It is sufficient to exclude the case $A=N$. We argue by contradiction and assume $A=N$. The commutator subgroup of $S_4$ is $A_4$ and therefore $A_4 < G' < \tilde G < G$. Since $\tilde G$ is known to be of order 96, it follows that $\tilde G = N \rtimes A_4$. We consider the isotropy group $I_1 \cong C_2 \times C_2 < \tilde G$ and its intersection with $A=N$.

If $A\cap I_1 = \{ \mathrm{id} \}$, then $A =N \cong C_2 ^3$ acts effectively on $\pi_1 (Y) =\mathbb P_1$, a contradiction. If $A \cap I_1 = I_1$, i.e., if $I_1$ is contained in $A$, then the quotient group $\tilde G / I_1$, which acts effectively on $\mathbb P_1$ by definition, is isomorphic to $C_2 \rtimes A_4 \cong C_2 \times A_4$. This group does however not admit an effective action on $\mathbb P_1$.

It remains to consider the case $A \cap I_1 \cong C_2$. 
The only nontrivial normal subgroup of $A_4$ is isomorphic to $C_2 \times C_2$. Since $I_1 \cong C_2 \times C_2$ is not contained in $A_4$ in the case under consideration, it follows that $I_1 \cap A_4 = \{ \mathrm{id}\}$.
Written as a subgroup of the semi-direct product $\tilde G = N \rtimes A_4$ the group $I_1 \cong C_2 \times C_2$ is of the form
$I_1 = \{ (\mathrm{id},\mathrm{id} ), (\sigma_1,\mathrm{id}), (\sigma_2, a), (\sigma_3, a)\}$
for $\sigma_i \in N$ and $a \in A_4$.
Using the fact that $I_1$ is a normal subgroup of $\tilde G$ one finds that $g a g^{-1} = a$ for each $g \in A_4 < \tilde G$, a contradiction.
\end{proof}
By the lemma above, the set $N\backslash A$ consists of four elements. These are involutions in $G$ not respecting the product structure of $Y \cong \mathbb P_1 \times \mathbb P_1$. In particular, each element of $N\backslash A$ exchanges the factors of $Y$.
Since both $N$ and $A$ are normal subgroup of $G$, it follows that $G$ acts on $N$, $A$, and also $N\backslash A$ by conjugation.
We consider an element $\lambda$ of order three contained in $A_4$, the commutator of the chosen copy of $S_4$ inside $G$. In particular, 
$\lambda \in A_4 = S_4' < G' < \tilde G < G$.
The action of $\lambda$ on the set $N\backslash A$ has at least one fixed point, i.e., there exists an element $\sigma \in N\backslash A$ such that $\lambda \sigma \lambda^{-1} = \sigma$.

After a suitable change of coordinates, the action of $\sigma$ on $Y$ is of the form $\sigma (z,w) = (w,z)$. Since $\lambda \in \tilde G$ respects the product structure and commutes with $\sigma$, it follows that
$\lambda (z,w) = (\lambda_z z, \lambda _z w)$.
for some $\lambda_z \in \mathrm{PSL}_2(\mathbb C)$ of order three. We choose homogeneous coordinates on $\pi_i(Y) \cong \mathbb P_1$ such that
$\lambda (z,w) = \lambda ([z_0: z_1], [w_0: w_1]) = ([ \xi z_0: z_1], [\xi w_0: w_1])$,
where $\xi$ is some nontrivial third root of unity. Note that this choice of coordinates does not effect the chosen shape of $\sigma$.

The branch curve $B$ of the covering $\pi: X \to Y$ is given by a $G$-(semi)-invariant polynomial $f$ of bidegree $(4,4)$. Since $\lambda$ is contained in the commutator of $G$, the polynomial $f$ is invariant with respect to the induced action of $\lambda$. In particular, it is a linear combination of $\lambda$-invariant monomials of bidegree $(4,4)$. These are
\[
z_0^4 w_0^2 w_1^2, \, \,
z_0^3 z_1 w_0^3 w_1,\, \,
z_0^3 z_1 w_1^4,\, \,
z_0^2 z_1^2 w_0^4,\, \,
z_0^2 z_1^2 w_0 w_1^3,\, \,
z_0 z_1^3 w_0^2 w_1^2,\, \,
z_1^4 w_0^3 w_1,\, \,
z_1^4 w_1^4.
\]
It follows that three of the four $\lambda$-fixed points on $Y$, namely 
$([1:0],[1:0])$, $([0:1][1:0])$, and  $([1:0],[0:1])$ lie on $B$ and the action of $\lambda$ on the double covering $X$ has at most five fixed points. However by Remark \ref{order of symp aut}, a symplectic action of $C_3$ on the K3-surface $X$ has precisely six fixed points. This yields a contradiction and we conclude $Y \not \cong \mathbb P_1 \times \mathbb P_1$.

As we have obtained contradictions for all possible choices of Del Pezzo surfaces $Y$ the non-existence claimed in Proposition \ref{192} follows. 
\subsection{The group \texorpdfstring{$N_{72} = C_3^2 \rtimes D_8$}{N72}}
We let $X$ be a K3-surface with a symplectic action of $G=N_{72}$ centralized by an antisymplectic involution $\sigma$ with $\mathrm{Fix}_X(\sigma) \neq \emptyset$.
Note that in this case we may not apply Theorem \ref{roughclassi} and therefore begin by excluding that a $G$-minimal model of $Y=X/\sigma$ is an equivariant conic bundle. Then, after limiting the possible degrees of $G$-minimal models $Y_\mathrm{min}$, we study the possibility of rational branch curves.
\begin{lemma}\label{N72notconicbundle}
Any $G$-minimal model of $Y$ is a Del Pezzo surface.
\end{lemma}
\begin{proof}
Assume the contrary and let $Y_\mathrm{min}$ be an equivariant conic bundle and a $G$-minimal model of $Y$.  We consider the induced action of $G$ on the base $\mathbb P_1$
and denote by $I \lhd G$ the ineffectivity of this action. Arguing as in the proof of Lemma \ref{conicbundle}, we see that $I$ is trivial or isomorphic to either $C_2$ or $C_2 \times C_2$. In all cases the quotient $G/I$ contains the subgroup $C_3 \times C_3$, which has no effective action on $\mathbb{P}_1$.
\end{proof}
\begin{lemma}
Let $Y_\mathrm{min}$ be a Del Pezzo surface and a $G$-minimal model of $Y$. Then $\mathrm{deg}(Y_\mathrm{min}) \leq 4$.
\end{lemma}
\begin{proof}
We exclude all $G$-minimal Del Pezzo surfaces of degree at least five:

A Del Pezzo surface of degree five has automorphism group $S_5$ (cf.\ Theorem 10.2.2 in \cite{dolgachev}). Since $N_{72} \nless S_5$, it follows that $\mathrm{deg}(Y_\mathrm{min}) \neq 5$

The automorphism group of a Del Pezzo surface of degree six is $(\mathbb C^* )^2 \rtimes (S_3 \times C_2)$ (cf.\ Theorem 10.2.1 in \cite{dolgachev}). Assume that $N_{72} = C_3^2 \rtimes D_8$ is contained in this group and consider the intersection $ A = N_{72} \cap (\mathbb C^* )^2$. The quotient of $N_{72}$ by $A$ is a subgroup of $S_3 \times C_2$ and may not contain a copy of $C_3^2$. Therefore, the order of $A$ is at least six and $A$ contains a copy of $C_3$. If $|A| =6$, then $A = C_6 = C_3 \times C_2$ and $C_2$ is central in $N_{72}$. Using explicitly the group structure of $N_{72}$ one finds however that there is no copy of $C_2$ in $N_{72}$ centralizing $C_3 \times C_3$.
If $|A|  >6$, then the centralizer of $C_3$ in $D_8$ has order greater than 2. This is contrary to the fact that for every choice of $C_3$ inside $C_3 \times C_3$ the centralizer inside $D_8$ is isomorphic to $C_2$. It follows that $\mathrm{deg}(Y_\mathrm{min}) \neq 6$.

If $G$ acts on $\mathbb P_1 \times \mathbb P_1$, then the canonical projections are equivariant with respect to a subgroup $H$ of index two in $G$. We consider one of these projections. The action of $H$ induces an effective action of $H/I$ on the base $\mathbb P_1$, where the group $I$ is either trivial or isomorphic to $C_2$ or $C_2 \times C_2$. In all case we find an effective action of $C_3 ^2$ on the base, a contradiction.  

It remains to exclude $\mathbb P_2$.
If $N_{72}$ acts on $\mathbb P_2$ we consider its embedding into $\mathrm{PSL}_3(\mathbb C)$, in particular the realization of the subgroup $C_3^2= \langle a \rangle \times \langle b \rangle$ 
and its lifting to $\mathrm{SL}_3(\mathbb C)$. 
One uses explicit realizations of the generators $a$ and $b$ in appropriately chosen coordinates and checks that the action of $D_8$ on $C_3^2$ cannot be realized in $\mathrm{PSL}_3(\mathbb C)$. The calculation is omitted here and the reader is referred to Section 4.8 in \cite{kf} for details.
It follows that there is no action of $N_{72}$ on $\mathbb P_2$.
\end{proof}
\begin{lemma}\label{noratN72}
There are no rational curves in $\mathrm{Fix}_X(\sigma)$.
\end{lemma}
\begin{proof}
 Let $n$ denote the total number of rational curves in $\mathrm{Fix}_X(\sigma)$ and recall $n \leq 10$. If $n \neq 0$, let $C$ be a rational curve in the image of  $\mathrm{Fix}_X(\sigma)$ in $Y$ and let $H = \mathrm{Stab}_G(C)$ be its stabilizer. 
Its order is at least eight and its action on $C$ is effective. 
First note that $G$ does not contain $S_4 = O_{24}$ as a subgroup. If this were the case, consider the intersection $S_4 \cap C_3^2$ and the quotient $S_4 \to S_4 / (S_4 \cap C_3^2) < D_8$. Since the only nontrivial normal subgroups of $S_4$ are $A_4$ and $C_2 \times C_2$, this leads to a contradiction.
Consequently, the order of $H$ is at most twelve. In particular, $n \geq 6$. We will obtain a contradiction by showing $n \leq 4$.

Since $| H| \geq 8$ and $C_8 \nless G$, the group $H$ is not cyclic and any $H$-orbit on $C$ consists of at least two points.
It follows from $C^2 = -4$ that $C$ must meet the union of Mori fibers and the union of Mori fibers meets the curve $C$ in at least two points. Recalling that each Mori fibers meets the branch locus $B$ in at most two points we see that at least $n$ Mori fibers meeting $B$ are required. However, no configuration of $n$ Mori fibers is sufficient to transform the curve $C$ into a curve on a Del Pezzo surface and further Mori fibers are required. By invariance, the total number $m$ of Mori fibers must be at least $2n$.
Combining the Euler-characteristic formula \eqref{eulerchar}
with our observation $\mathrm{deg}(Y_\mathrm{min}) \leq 4$, i.e., $e(Y_\mathrm{min}) \geq 8$ we see that $n \leq 4$.
\end{proof}
\begin{proposition}
Let $X$ be a K3-surface with a symplectic action of $G=N_{72}$ centralized by an antisymplectic involution $\sigma$ with $\mathrm{Fix}_X(\sigma) \neq \emptyset$.
 Then the quotient surface $Y = X / \sigma $ is $G$-minimal and isomorphic to the Fermat cubic $\{x_1^3 + x_2^3 +x_3^3 +x_4^3 =0\} \subset \mathbb P_3$. Up to equivalence, there is a unique action of $G$ on $Y$ and the branch locus of $X \to Y$ is given by $\{x_1x_2 + x_3x_4 =0\}$. In particular, $X$ is equivariantly isomorphic to Mukai's $N_{72}$-example.
\end{proposition}
\begin{proof}
Using the Euler-cha\-rac\-te\-ris\-tic formula \eqref{eulerchar} together with the two previous lemmata, we find that the total number $m$ of Mori fibers is bounded by four. Since the maximal order of a stabilizer group of a Mori fiber is twelve (cf.\ proof of Theorem \ref{roughclassi}) we conclude $m=0$, i.e., $Y$ must be $G$-minimal.

In order to conclude that $Y$ is the Fermat cubic we consult Dolgachev's lists of automorphisms groups of Del Pezzo surfaces of degree less than or equal to four (\cite{dolgachev} Section 10.2.2; Tables 10.3; 10,4; and 10.5):
It follows immediately from the order of $G$ that $Y$ is not of degree two or four. If $G$ were a subgroup of an automorphism group of a Del Pezzo surface of degree one, it would contain a central copy of $C_3$, which is not the case.
After excluding the cases $\mathrm{deg}(Y) \in \{1,2,4\}$
it remains to consider the case of a cubic hypersurface $Y$ in $\mathbb P_3$. 
The action of $G$ on $Y$ is induced by a four-dimensional (projective) representation of $G$ and the branch curve $C \subset Y$ is the intersection of $Y$ with an invariant quadric (compare proof of Proposition \ref{S5 on degree three}). It follows from the representation and invariant theory of the group $N_{72}$ that there is a unique action of $N_{72}$ on $\mathbb P_3$, a unique invariant cubic hypersurface, namely the Fermat cubic, and a unique invariant quadric hypersurface $\{x_1x_2 + x_3x_4 =0\}$. The necessary computations are carried out in Appendix A.1 in \cite{kf}.

Mukai's $N_{72}$-example is defined by $\{ x_1^3+ x_2 ^3 + x_3^3 +x_4^3= x_1x_2 + x_3x_4+ x_5^2 = 0 \} \subset \mathbb P_4$. An anti-symplectic involution centralizing the action of $N_{72}$ is given by the map $ x_5 \mapsto -x_5$. The quotient of Mukai's example by this involution is the Fermat cubic and the fixed point set of the involution is given by $\{x_1x_2 + x_3x_4= 0 \}$.
\end{proof}
\subsection{The group \texorpdfstring{$M_9 =C_3^2 \rtimes Q_8$}{M9}}
Let $G = M_9$ and let $X$ be a K3-surface with a symplectic $G$-action centralized by the antisymplectic involution $\sigma$ such that $\mathrm{Fix}_X(\sigma) \neq \emptyset$.
We proceed in analogy to the case $G=N_{72}$ above. Arguing precisely as in the proof of Lemma \ref{N72notconicbundle} one shows that any $G$-minimal model of $Y$ is a Del Pezzo surface. As a next step we exclude rational branch curves.
\begin{lemma}\label{subgroupsM9}
There are no rational curves in $\mathrm{Fix}_X(\sigma)$.
\end{lemma}
\begin{proof}
Let $n$ be the total number of rational curves in $\mathrm{Fix}_X(\sigma)$. Assume $n \neq 0$, let $C$ be a rational curve in the image of $\mathrm{Fix}_X(\sigma)$ in $Y$ and let $H <G$ be its stabilizer. The action of $H$ on $C$ is effective. We go through the list of finite groups with an effective action on a rational curve. 
Since $M_9$ is a group of symplectic transformations on a K3-surface, its element have order at most eight.
Clearly, $A_5, \, D_{10}, \, D_{14}, \, D_{16} \nless M_9$. If $S_4 < M_9 = C_3^2 \rtimes Q_8$, then $S_4 \cap C_3^2$ is a normal subgroup of $S_4$ and it is therefore trivial. Now $ S_4 = S_4 / (S_4 \cap C_3^2) <  M_9 / C_3^2 =  Q_8$ yields a contradiction. The same argument can be carried out for $A_4$, $D_8$ and $C_8$.  If $D_{12} < M_9 = C_3^2 \rtimes Q_8$, then either $D_{12} \cap C_3^2 = C_3$ and $C_2 \times C_2 = D_{12} / C_3 < M_9 / C_3^2 =  Q_8$ or $D_{12} \cap C_3^2 = \{ \mathrm{id} \}$ and $D_{12} < Q_8$, both are impossible.
It follows that the subgroups of $M_9$ admitting an effective action on a rational curve have index greater than or equal to twelve. Therefore  $n \geq 12$, contrary to inequality \eqref{atmostten} stating $n \leq 10$.
\end{proof}
\begin{proposition}
 The quotient surface $Y$ is $G$-minimal and isomorphic to $\mathbb P_2$. Up to equivalence, there is a unique action of $G$ on $Y$ and the branch locus of $X \to Y$ is given by
$\{x_1^6 + x_2^6+ x_3^6-10( x_1^3x_2^3 + x_2^3x_3^3+ x_3^3x_1^3 ) =0\}$. In particular, $X$ is equivariantly isomorphic to Mukai's $M_9$-example.
\end{proposition}
\begin{proof}
We first check that $Y$ is $G$-minimal. Again, we proceed as in the proof of Theorem \ref{roughclassi} and Lemma \ref{subgroupsM9} above to see a stabilizer group of a Mori fiber has order at most six. If $Y$ is not $G$-minimal, this implies that the total number of Mori fibers is $ \geq 12$, contradicting $m \leq 9$.
Note that $X \to Y$ is not branched along one or two elliptic curves as this would imply $e(Y) =12$ and contradict the fact that $Y$ is a Del Pezzo surface. 

Let $B$ be the branch curve of $ X \to  Y$ and consider the quotient $Q$ of $B$ by the normal subgroup $N = C_3^2$ in $G$. On $Q$ there is an action of $Q_8$ implying that $Q$ is not rational or elliptic.
 It follows that the genus of $Q$ is at least two and the genus of $B$ is at least ten. Adjunction on the Del Pezzo surface $Y$ now implies $g=10$ and $Y \cong \mathbb P_2$. 

It follows from direct computation involving the generators of $M_9$ (cf.\ Appendix A.2 in \cite{kf}) that,
up to natural equivalence, there is a unique action of $M_9$ on the projective plane. We may therefore consider the explicit action of $M_9$ on $\mathbb P_2$ specified by Mukai (cf.\ \cite{mukai}).
Studying the induced action of $M_9$ on the space of sextic curves
one finds three $M_9$-invariant sextic curves, namely
$\{x_1^6 + x_2^6+ x_3^6-10( x_1^3x_2^3 + x_2^3x_3^3+ x_3^3x_1^3 ) =0\}$, 
which is the example presented by Mukai, and additionally the two curves defined by 
$f_a=x_1^6 + x_2^6+ x_3^6 + (18-3a)(x_1^2x_2^2x_3^2) +2(x_1^3x_2^3 + x_1^3x_3^3+ x_2^3x_3^3)
      + a(x_1^4x_2x_3 + x_1x_2^4x_3+ x_1x_2x_3^4)$, 
where $a$ is a solution of the quadratic equation $a^2-6a+36$.
Depending on the choice of $a$, the polynomial $f_a$ is either invariant or semi-invariant with respect to the action of $M_9$.

We need to prove that $X$ is not the double cover of $\mathbb P_2$ branched along $\{f_a=0\}$. If this was the case consider the fixed point $p=[0:1:-1]$ of the automorphism $I \in Q_8 < M_9 < \mathrm{PSL}_3(\mathbb C)$ given by
\[
\tilde I= \frac{1}{\xi -\xi^2}
 \begin{pmatrix}
1 & 1 & 1 \\
1 & \xi & \xi^2\\
1 & \xi^2& \xi
 \end{pmatrix} \in \mathrm{SL}_3(\mathbb C)
\]
for some third root of unity $\xi$
and note that $f_a(p)=0$. In particular, the fiber $\pi^{-1}(p)$ consists of one point $x \in X$. We linearize the $\langle I \rangle \times \langle \sigma \rangle$-action at $x$. In suitably chosen coordinates the action of the symplectic automorphism $I$ of order four is of the form  $(z,w) \mapsto (iz, -iw)$. Since the action of $\sigma$ commutes with $I$, the $\sigma$-quotient of $X$ is locally given by
$(z,w) \mapsto (z^2, w)$ or $(z,w) \mapsto (z, w^2)$.
It follows that the action of $I$ on $Y$ is locally given by either
$ (x,y) \mapsto (-x, -i y)$ or
$ (x, y) \mapsto (i x, -y)$.
In particular, the local linearization of $I$ at $p$ has determinant $\neq 1$. 
By a direct computation using the explicit form $\tilde I \in \mathrm{SL}_3(\mathbb C)$ of the generator $I$, in particular the facts that $\mathrm{det}(\tilde I) =1$ and $\tilde I v =v$ for $[v]=p$, we obtain a contradiction.
 This completes the proof of the proposition.
\end{proof}
\begin{remark}\label{M9 symplectic}
In the proof of the proposition above we have observed that an element of $\mathrm{SL}_3(\mathbb C)$ does not necessarily lift to a symplectic transformation on the double cover of $\mathbb P_2$ branched along a sextic given by an invariant polynomial.
Mukai's $M_9$-example $X$ is the double cover of $\mathbb P_2$ branched along the sextic curve $\{x_1^6 + x_2^6+ x_3^6-10( x_1^3x_2^3 + x_2^3x_3^3+ x_3^3x_1^3 ) =0\}$ and for this particular example, the action of $M_9$ does lift to a group of symplectic transformation as stated by Mukai.\\
To see this consider the set $\{a,b,I,J\}$ of generators of $M_9 =C_3^2 \rtimes Q_8= (\langle a \rangle  \times \langle b \rangle  ) \rtimes Q_8$. Since $a$ and $b$ are commutators in $M_9$, they can be lifted to symplectic transformation $\overline a, \overline b$ on $X$. For $I,J$ consider the linearization at the fixed point $[0:1:-1] \in \mathbb P_2$ and, using the explicit realization of the group, one checks that it has determinant one. Since $[0:1:-1]$ is \emph{not} contained in the branch set of the covering, its preimage in $X$ consists of two points $p_1,p_2$. We can lift $I$ ($J$, respectively) to a transformation of $X$ fixing both $p_1,p_2$ and a neighbourhood of $p_1$ is $I$-equivariantly isomorphic to a neighbourhood of $ [0:1:-1] \in \mathbb P_2$. In particular, the action of the lifted element $\overline I$ ($\overline J$, respectively) is symplectic. On $X$ there is the action of a degree two central extension $E$ of $M_9$.
The elements $\overline a, \overline b, \overline I, \overline J$ generate a subgroup $\tilde M_9$ of $E_\mathrm{symp}$ mapping onto $M_9$. Since $E_\mathrm{symp} \neq E$, the order of $\tilde M_9$ is 72 and it follows that $\tilde M_9$ is isomorphic to $M_9$. In particular $E$ splits as $E_\mathrm{symp} \times C_2$ with  $E_\mathrm{symp}= M_9$.
\end{remark}
\subsection{The group \texorpdfstring{$T_{48} = Q_8 \rtimes S_3$}{T48}}
We begin by specifying the group structure of $T_{48}$: the element $c$ of order three in $S_3$ acts on $Q_8$ by permuting $I,J,K$ and an element $d$ of order two acts by exchanging $I$ and $J$ and mapping $K$ to $-K$.
Now let $X$ be a K3-surface with an action of $T_{48} \times C_2$ where the action of $G = T_{48}$ is symplectic and the generator $\sigma$ of $C_2$ is antisymplectic and has fixed points. 
\begin{lemma}\label{not two}
A $G$-minimal model $Y_\mathrm{min}$ of $Y$ is either $\mathbb P_2$, a Hirzebruch surface $\Sigma_k$ with $k >2$, or $e(Y_\mathrm{min}) \geq 9$.
\end{lemma}
\begin{proof}
Let us first consider the case where $ Y_\mathrm{min}$ is a Del Pezzo surface.

If $Y_\mathrm{min} \cong \mathbb P_1 \times \mathbb P_1$, then both canonical projections are equivariant with respect to the index two subgroup $G'= Q_8 \rtimes C_3$. 
Since $Q_8$ has no effective action on $\mathbb P_1$, it follows that the subgroup $Z =\{+1,-1\} < Q_8$ acts trivially on the base. This holds with respect to both projections and the subgroup $Z$ acts trivially on $Y_\mathrm{min}$, a contradiction. 

Using the group structure of $T_{48}$ one checks that the only nontrivial normal subgroup $N$ of $T_{48}$ such that $N \cap Q_8 \neq Q_8$ is the center $Z= \{+1,-1\}$ of $T_{48}$. It follows that $T_{48}$ is neither a subgroup of $(\mathbb C^*)^2 \rtimes (S_3 \times C_2)$ nor a subgroup of any of the automorphism groups $C_2^4 \rtimes \Gamma$ for $\Gamma \in \{C_2, C_4, S_3, D_{10}\}$ of a Del Pezzo surface of degree four. 
Furthermore,
$T_ {48} \nless S_5$ and consequently $\mathrm{deg}(Y_\mathrm{min}) \neq 4,5,6$.

So if $Y_\mathrm{min}$ is a Del Pezzo surface, then $Y_\mathrm{min} \cong \mathbb P_2$ or $e(Y_\mathrm{min}) \geq 9$ .

Let us now turn to the case where $Y_\mathrm{min}\to \mathbb P_1$ is an equivariant conic bundle.
The center $Z = \{+1,-1\}$ of $G= T_{48}$ acts trivially on the base and has two fixed points in the generic fiber. Let $\Lambda_1$ and $\Lambda_2$ denote the two curves of $Z$-fixed points in $Y_\mathrm{min}$. 

We first show that $Y_\mathrm{min}$ is not a conic bundle with singular fibers.
Any singular fiber $F$ is the union of two (-1)-curves $F_1,F_2$ meeting transversally in one point. We consider the action of $Z$ on this union of curves. The group $Z$ does not act trivially on either component of $F$ since linearization at a smooth point of $F$ would yield a trivial action of $Z$ on $Y_\mathrm{min}$.
Consequently,
it has either one or three fixed points on $F$. 
The first is impossible since $\Lambda_1$ and $\Lambda_2$ intersect $F$ in two points. It follows that $Z$ stabilizes each curve $F_i$. We linearize the action of $Z$ at the point of intersection $F_1 \cap F_2$. The intersection is transversal and the action of $Z$ is by $-\mathrm{Id}$ on the tangent space $T_{F_1} \oplus T_{F_2}$ contradicting the fact the $Z$ acts trivially on the base. Thus $Y_\mathrm{min}$ is not a conic bundle with singular fibers.

It remains to consider the case where $p:Y_\mathrm{min} = \Sigma _k \to \mathbb P_1$ is a Hirzebruch surface.
The curves $\Lambda_1$ and $\Lambda_2$ are disjoint sections of $p$. 
This is only possible if $\Lambda_1^2 = -\Lambda_2^2 = k$. 
In particular, the action of $T_{48}$ on $\Sigma_k$ stabilizes each curve $\Lambda_i$.
Removing the exceptional section $\Lambda_2= E_\infty$ from $\Sigma_k$, we obtain $H^k$, the $k^\text{th}$ power of the hyperplane bundle of $\mathbb P_1$. We chose the section $\Lambda_2$ to be the zero section and conclude that the action of $T_{48}$ on $H^k$ is by bundle automorphisms. 
If $k=2$, then $H^k$ is the anticanonical line bundle of $\mathbb P_1$ and the effective action of $S_4$ on the base induces an action of $S_4$ on $H^2$ by bundle automorphisms. It follows that $T_{48}$ splits as $S_4 \times C_2$, a contradiction. Thus, if $Y_\mathrm{min}$ is a Hirzebruch surface $\Sigma_k$, then $k \neq 2$. 
\end{proof}
\begin{lemma}\label{no rat T48}
 There are no rational curves in $\mathrm{Fix}_X(\sigma)$.
\end{lemma}
\begin{proof}
We let $n$ denote the total number of rational curves in $\mathrm{Fix}_X(\sigma)$ and assume $n >0$. Recall $n \leq 10$, let $C$ be a rational curve in $B=\pi(\mathrm{Fix}_X(\sigma)) \subset Y$ and let $H = \mathrm{Stab}_G(C) < G$ be its stabilizer group. The action of $H$ on $C$ is effective, the index of $H$ in $G$ is at most 8. Using the quotient homomorphism $T_{48} \to T_{48}/Q_8 = S_3$ one checks that $T_{48}$ does not contain $O_{24}=S_4$ or $T_{12}= A_4$ as a subgroup. It follows that $H$ is a cyclic or a dihedral group.

If $H\in \{C_6, C_8, D_8\}$, then $H$ and all conjugates of $H$ in $G$ contain the center $Z= \{+1,-1\}$ of $G$. It follows that $Z$ has two fixed point on each curve $gC$ for $g \in G$. Since there are six (or eight) distinct curves $gC$ in $Y$, it follows that $Z$ has at least 12 fixed points in $Y$ and in $X$. This contradicts to assumption that $Z < G$ acts symplectically on $X$ and therefore has eight fixed points in the K3-surface $X$.

It remains to study the cases $H = D_{12}$, i.e., $n=4$, and $H = D_6$, i.e., $n = 8$. 

We note that a Hirzebruch surface has precisely one curve with negative self-intersection and only fibers have self-intersection zero. A Del Pezzo surface does not contains curves of self-intersection less than $-1$. The rational branch curves must therefore meet the union of Mori fibers in $Y$.

The total number of Mori fibers is bounded by $n+9$. We study the possible stabilizer subgroups $\mathrm{Stab}_G(E) < G$ of Mori fibers. A Mori fiber $E$ with self-intersection (-1) meets the branch locus $B$ in one or two points and its stabilizer is either cyclic or dihedral. If $\mathrm{Stab}_G(E) \in \{C_4, D_8\}$, then the points of intersection of $E$ and $B$ are fixed points of the center $Z$ of $G$ and we find too many $Z$-fixed points on $X$. 

Assume $n=4$ and let $R_1, \dots R_4$ be the rational curves in $B$. We denote by $\tilde R_i$ their images in $Y_\mathrm{min}$. The total number $m$ of Mori fibers is bounded by 12.
We go through the list of possible configurations:
\begin{itemize}
 \item 
If $m = 4$, there is no invariant configuration of Mori fibers such that the contraction maps the four rational branch curves to an admissible configuration on the Hirzebruch or Del Pezzo surface $Y_\mathrm{min}$.
\item
If $m= 6$, then $\mathrm{Stab}_G(E) = C_8$ and the points of intersection of $E$ and $B$ are $Z$-fixed. Since $Z$ has at most eight fixed points on $B$, it follows that each curve $E$ meets $B$ only once. The images $\tilde R_i$ of the $R_i$ contradict our observations about curves in Del Pezzo and Hirzebruch surfaces.
\item
If $m=8$ and all Mori fibers have self-intersection $-1$, then each Mori fiber meets $\bigcup R_i$ in a $Z$-fixed point.
 Since there at at most eight such points, it follows that each Mori fibers meets $\bigcup R_i$ only once and their contractions does not transform the curves $R_i$ sufficiently. 
\item
If $m=8$ and only four Mori fibers have self-intersection $-1$, we consider the four Mori fibers of the second reduction step. Each of these meets a Mori fiber $E$ of the first step in precisely one point. By invariance, this would have to be a fixed points of the stabilizer $ \mathrm{Stab}_G(E)= D_{12}$, a contradiction.
\item
If $m=12$, then either $e(Y_\mathrm{min}) = 3$ and there exist a branch curve $B_{g=2}$ of genus two or $e(Y_\mathrm{min}) = 4$ and $B = \bigcup R_i$. In the first case, $Y_\mathrm{min}  \cong \mathbb P_2$ and twelve Mori fibers are not sufficient to transform $B = B_{g=2} \cup \bigcup R_i$ into an admissible configuration of curves in the projective plane.
So $Y_\mathrm{min} = \Sigma_k$ for $k > 2$.
Recall that $Z$ has two fixed points in each fiber of $p: \Sigma_k \to \mathbb P_1$, i.e., the $Z$-action on $\Sigma_k$ has two disjoint curves of fixed points. As was remarked above, these curves are the exceptional section $E_\infty$ of self-intersection $-k$ and a section $E_0 \sim E_\infty + k F$ of self-intersection $k$. Here $F$ denotes a fiber of $p: \Sigma_k \to \mathbb P_1$. There is no automorphisms of $\Sigma_k$ mapping $E_\infty$ to $E_0$.
Each rational branch curve $\tilde R_i$ has two $Z$-fixed points. These are exchanged by an element of $\mathrm{Stab}_G(R_i)$ and therefore both lie on either $E_\infty$ or $E_0$, i.e., $\tilde R_i$ cannot have nontrivial intersection with both $E_0$ and $E_\infty$. By invariance all curves $\tilde R_i$ either meet $E_0$ or $E_\infty$ and not both.
Using the fact that $\sum \tilde R_i$ is linearly equivalent to $-2K_{\Sigma_k} \sim 4 E_\infty +(2k +4)F$ we find that $\tilde R_i \cdot E_\infty = 0$ and $k=2$, a contradiction to Lemma \ref{not two}. 
\end{itemize}
We have shown that
all possible configurations in the case $n \neq 4$ lead to a contradiction. We now turn to the case $n=8$ and let $R_1, \dots R_8$ be the rational branch curves. The total number of Mori fibers is bounded by 16. Note that by invariance, the $G$-orbit of a Mori fiber meets $\bigcup R_i$ in at least 16 points or not at all. In particular, Mori fibers meeting $R_i$ come in orbits of length $\geq 8$. As above, we go through the list of possible configurations.
\begin{itemize}
 \item 
If $m =16$, then the set of all Mori fibers consists of one orbit of length 16 or of two orbits of length eight. If all 16 Mori fibers meet $B$, then each meets $B$ in one point and $R_i$ is mapped to a (-2)-curve in $Y_\mathrm{min}$. If only eight Mori fibers meet $B$, then each of the eight Mori fibers of the second reduction step meets one Mori fiber $E$ of the first reduction step in one point. This point has to be a $\mathrm{Stab}_G(E)$-fixed point. The $\mathrm{Stab}_G(E)$-fixed points on $E$ must however coincide with the points $E \cap B$, a contradiction.
\item
If $m=12$, then the set of all Mori fibers consists of a single $G$-orbit and each curve $R_i$ meets three distinct Mori fibers. Their contraction transforms $R_i$ into a (-1)-curve on $Y_\mathrm{min}$. It follows that $Y_\mathrm{min}$ contains at least eight (-1)-curves and is a Del Pezzo surface of degree $\leq 5$. We have seen above that $\mathrm{deg}( Y_\mathrm{min}) \neq 4,5$ and therefore $e(Y_\mathrm{min}) \geq 9$. With $m=12$ and $n=8$, this contradicts the Euler characteristic formula $24 = 2 e(Y_\mathrm{min}) +2m -2n +(2g-2)$.
\item
If $m=8$ there is no invariant configuration of Mori fibers such that the contraction maps the eight rational branch curves to an admissible configuration on the Hirzebruch or Del Pezzo surface $Y_\mathrm{min}$.
\end{itemize}
This completes the proof of the lemma.
\end{proof}
Since there is an effective action of $T_{48}$ on $\mathrm{Fix}_X(\sigma)$, it is neither an elliptic curve nor the union of two elliptic curves. It follows that $ X \to Y$ is branched along a single $T_{48}$-invariant curve $B$ with $g(B) \geq 2$.
\begin{lemma}
The genus of $B$ is neither three nor four.
\end{lemma}
\begin{proof}
We consider the quotient $Q = B/Z$ of the curve $B$ by the center $Z$ of $G$ and apply the Euler characteristic formula,  $e(B) = 2 e(Q) - |\mathrm{Fix}_B(Z)|$. On $Q$ there is an effective action of the group $G/Z = (C_2 \times C_2) \rtimes C_3 = S_4$. Using Remark \ref{S4 not on g=1,2} we see that $e(Q) \in \{2, -4, -6,-8 \dots\}$.

If $g(B)=3$, then $e(B) = -4$ and the only possibility is $Q \cong \mathbb P_1$ and $|\mathrm{Fix}_B(Z)| =8$. In particular, all $Z$-fixed points on $X$ are contained in the curve $R = \pi^{-1}(B)$. Let $A < G$ be the group generated by $I \in Q_8 = \{ \pm 1, \pm I, \pm J, \pm K \}$. The four fixed points of $A$ in $X$ are contained in $\mathrm{Fix}_X(Z) = \mathrm{Fix}_B(Z)$ and the quotient group $A/Z \cong C_2$ has four fixed points in $Q$. This is a contradiction.

If $g(B)=4$, then $e(B) = -6$ and the only possibility is $Q \cong \mathbb P_1$ and $|\mathrm{Fix}_B(Z)| =10$. This contradicts the fact that $Z$ has at most eight fixed points in $B$ since it has precisely eight fixed points in $X$.
\end{proof}
In Lemma \ref{not two} we have reduced the classification to the cases $e(Y_\mathrm{min}) \in \{3,4, 9, 10, 11\}$. In the following, we will exclude the cases $e(Y_\mathrm{min}) \in \{4, 9, 10\}$ and describe the remaining cases more precisely. Recall that the maximal possible stabilizer subgroup of a Mori fiber is $D_{12}$, in particular, $m = 0$ or $ m \geq 4$.
\begin{lemma}
 If  $e(Y_\mathrm{min}) =3$, then $Y_\mathrm{min} = Y = \mathbb P_2$ and $X \to Y$ is branched along the curve $\{x_1x_2(x_1^4-x_2^4) + x_3^6=0\}$. In particular, $Y$ is equivariantly isomorphic to Mukai's $T_{48}$-example.
\end{lemma}
\begin{proof}
Let $ M: Y \to \mathbb P_2$ denote a Mori reduction of $Y$ and let $B \subset Y$ be the branch curve of the covering $X \to Y$. 
If $Y = Y_\mathrm{min}$, then $B= M(B)$ is a smooth sextic curve.
If $Y \neq Y_\mathrm{min}$, then the Euler characteristic formula with $m \in \{4,6,8\}$ shows that $g(B) \in \{2,4,6\}$. The case $m=6$, $g(B) =4$ has been excluded by the previous lemma.

If $m=4$, then the stabilizer group of each Mori fiber is $D_{12}$ and each Mori fiber meets $B$ in two points. 
The image $M(B)$ of $B$ in $Y_\mathrm{min}$ is an irreducible singular sextic.

If $m=8$, then $g(B) =2$ and $B^2 =4$. Since the self-intersection number $M(B)^2$ must be a square, one checks that all possible invariant configurations of Mori fibers yield $M(B)^2 =36$ and involve Mori fibers meeting $B$ in two points. In particular, also in this case $M(B)$ must be a singular sextic.

By explicit computation using the group structure of $T_{48}$ one can determine the unique action of $T_{48}$ on $\mathbb P_2$. This is carried out in detail in Section 4.10 in \cite{kf}.
Using the explicit form of the $T_{48}$-action and the fact that the commutator subgroup of $T_{48}$ is $Q_8 \rtimes C_3$ one then checks that any invariant curve of degree six is of the form 
\[
C_\lambda = \{ x_1x_2(x_1^4-x_2^4) + \lambda x_3^6 =0\}
\]
for some $\lambda \in \mathbb C^*$.
In order to avoid this calculation, one can also argue that the polynomial $x_1x_2(x_1^4-x_2^4)$ is the lowest order invariant of the octahedral group $S_4 \cong T_{48} / Z$. 
The curve $C_\lambda$ is smooth and it follows that $Y = Y_\mathrm{min}$.
We may adjust the coordinates equivariantly such that $\lambda =1$ and find that our surface $X$ is precisely Mukai's $T_{48}$-example.
\end{proof}
\begin{remark}\label{T48 symplectic}
As stated by Mukai, the action of $T_{48}$ on $\mathbb P_2$ does indeed lift to a symplectic action of $T_{48}$ on the double cover of $\mathbb P_2$ branched along the invariant curve $\{ x_1x_2(x_1^4-x_2^4) +  x_3^6 =0\}$. The elements of the commutator subgroup can be lifted to symplectic transformation on the double cover $X$. 
The remaining generator $d$ 
is an involution fixing the point $[0:0:1]$. Any involution $\tau$ with a fixed point $p$ outside the branch locus can be lifted to a symplectic involution on the double cover $X$ as follows:
The linearized action of $\tau$ at $p$ has determinant $\pm 1$. We consider the lifting $\tilde \tau$ of $\tau$ fixing both points in the preimage of $p$. Its linearization coincides with the linearization on the base and therefore also has determinant $\pm 1$. 
In particular, $\tilde \tau$ is an involution. 
It follows that either $\tilde \tau$ or the second choice of a lifting $\sigma \tilde \tau$ acts symplectically on $X$. 
The group generated by all lifted automorphisms is either isomorphic to $T_{48}$ or to its degree two central extension $E$
acting on the double cover.
Since $E_\mathrm{symp} \neq E$ the later is impossible it follows that $E$ splits as $E_\mathrm{symp} \times C_2$ with $E_\mathrm{symp} = T_{48}$.
\end{remark}
Finally, we return to the remaining possibilities $e(Y_\mathrm{min}) \in \{4, 9, 10, 11\}$.
\begin{lemma}
$e(Y_\mathrm{min}) \not\in \{4,9,10\}$.
\end{lemma}
\begin{proof}
Recalling that the genus of the branch curve $B$ is neither three nor four and that $m$ is either zero or $\geq 4$, we may exclude $e(Y_\mathrm{min}) =9,10$ using the Euler characteristic formula $12 = e(Y_\mathrm{min}) +m +g-1$. It remains to consider the case $Y_\mathrm{min} = \Sigma _k$ with $k >2$ and we claim that this is impossible.

Let $M: Y \to Y_\mathrm{min} = \Sigma_k$ denote a (possibly trivial) Mori reduction of $Y$. The image $M(B)$ of $B$ in $\Sigma _k$ is linearly equivalent to $-2K_{\Sigma _k}$. Now $M(B) \cdot E_\infty = 2(2-k) < 0$ and it follows that $M(B)$ contains the rational curve $E_\infty$. This is a contradiction since $B$ does not contain any rational curves by Lemma \ref{no rat T48}.
\end{proof}
In the last remaining case, i.e., $e(Y_\mathrm{min})=11$, the quotient surface $Y$ is a $G$-minimal Del Pezzo surface of degree 1. Consulting \cite{dolgachev}, Table 10.5, we find that $Y$ is a hypersurface in weighted projective space $\mathbb P(1,1,2,3)$ defined by the weighted homogeneous equation
$x_1x_2(x_1^4-x_2^4)+ x_3^3+x_4^2$.
This follows from the invariant theory of the group $S_4 \cong T_{48}/Z$ and fact that the anticanonical map realizes  $Y$ as a double cover of a quadric cone $Q$ in $\mathbb P_3$ branched along the intersection of $Q$ with a cubic hypersurface.

The linear system of the anticanonical divisor $K_Y$ has precisely one base point $p$. In coordinates $[x_1:x_2:x_3:x_4]$ this point is given as $[0:0:1:i]$. It is fixed by the action of $T_{48}$. The linearization of $T_{48}$ at $p$ is given by the unique faithful 2-dimensional represention of $T_{48}$. 
It follows that there is a unique action of $T_{48}$ on $Y$. The branch curve $B$ is linearly equivalent to $-2K_Y$, i.e., $B = \{s=0\}$ for a section $s \in \Gamma(Y, \mathcal O(-2K_Y))$ which is either invariant or semi-invariant.

The adjunction formula for hypersurfaces in weighted projective space (cf.\ Theorem 3.3.4 in \cite{DolgWPS}) yields $\mathcal O (-2K_Y)) = \mathcal O _Y(2)$. The four-dimensional space of sections $\Gamma(Y, \mathcal O(-2K_Y))$ is generated by the weighted homogeneous polynomials $x_1^2, x_2^2, x_1x_2, x_3$. 
We consider the map $ Y \to \mathbb P (\Gamma(Y, \mathcal O(-2K_Y))^*)$ associated to $|-2K_Y|$. Since this map is equivariant with respect to $\mathrm{Aut}(Y)$, the fixed point $p$ is mapped to a fixed point in $\mathbb P (\Gamma(Y, \mathcal O(-2K_Y))^*)$. It follows that the section corresponding to the homogeneous polynomial $x_3$ is invariant or semi-invariant with respect to $T_{48}$. It is the only section of $\mathcal O(-2K_Y)$ with this property since the representation of $T_{48}$ on the span of $x_1^2, x_2^2, x_1x_2$ is irreducible.
It follows that $B = \{x_3 =0\}$.

In order to prove existence, let $b: Y \to \mathbb P_2$ be the blow down of eight disjoint (-1)-curves in $Y$. The curve $C = b(B)$ is a singular sextic. The double cover $X$ of $Y$ branched along $B$ is the minimal desingularization of the double cover $X_\mathrm{sing}$ of $\mathbb P_2$ branched along $C$. In particular, one finds that $X$ is a K3-surface. 
It remains to check that the action of $T_{48}$ on $Y$ lifts to a group of symplectic transformation on $X$. First note that $B$ does not contain the base point $p$. 
For $I,J,K, c \in T_{48}$ we choose liftings $ \overline I, \overline J, \overline K , \overline c \in \mathrm{Aut}(X)$ fixing both points in $\pi^{-1}(p)= \{p_1,p_2\}$. The linearization of $ \overline I, \overline J, \overline K , \overline c$ at $p_1$ is the same as the linearization at $p$ and in particular has determinant one. 
By the general considerations in Remark \ref{T48 symplectic} the involution $d$ can be lifted to a symplectic involution on $X$.
The symplectic liftings of $I,J,K,c,d$ generate a subgroup $\tilde G$ of $\mathrm{Aut}(X)$ which is isomorphic to either $T_{48}$ or to the central degree two extension of $T_{48}$ acting on $X$. 
In analogy to Remarks \ref{M9 symplectic} and \ref{T48 symplectic} we conclude that $\tilde G \cong T_{48}$ and the action of $T_{48}$ on $Y$ induces a symplectic action of $T_{48}$ on the double cover $X$.

This completes the classification of K3-surfaces with $T_{48} \times C_2$-symmetry. We have shown:
\begin{proposition}
 Let $X$ be a K3-surface with a symplectic action of the group $T_{48}$ centralized by an antisymplectic involution $\sigma$ with $\mathrm{Fix}_X(\sigma) \neq \emptyset$. Then $X$ is equivariantly isomorphic either to Mukai's $T_{48}$-example or to the double cover of 
$\{x_1x_2(x_1^4-x_2^4)+ x_3^3+x_4^2=0\} \subset \mathbb P(1,1,2,3)$
branched along $\{x_3=0\}$
\end{proposition}
\begin{remark}
The automorphism group of the Del Pezzo surface $Y = \{x_1x_2(x_1^4-x_2^4)+ x_3^3+x_4^2=0\} \subset \mathbb P(1,1,2,3)$ is the trivial central extension $C_3 \times T_{48}$.  By contruction, the curve $B=\{s=0\}$ is invariant with respect to the full automorphism group. The double cover $X$ of $Y$ branched along $B$ carries the action of a finite group $\tilde G$ of order $2 \cdot 3 \cdot 48 = 288$ containing $T_{48} < \tilde G_\mathrm{symp}$. Since $T_{48}$ is a maximal group of symplectic transformations, we find $T_{48} = \tilde G_\mathrm{symp}$.
In analogy to the proof of Claim 2.1 in \cite{OZ168}, one can check that 288 is the maximal order of a finite group $H$ acting on a K3-surface with $T_{48} < H_\mathrm{symp}$. It follows that $\tilde G$ is a maximal finite subgroup of $\mathrm{Aut}(X)$.
\end{remark}
\subsection*{Conclusion}
The classification and non-existence results for the individual groups from Table \ref{TableMukai} obtained in the preceding subsections yield the proof of Theorem \ref{thm mukai times invol}.
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