Let $\mathcal{S}$ be the set of permutations of $\{1,2,\ldots,6\}$, and let $\mathcal{T}$ be the set of permutations of $\mathcal{S}$ that preserve compositions: i.e., if $F\in\mathcal{T}$ then \[F(f_2\circ f_1)=F(f_2)\circ F(f_1)\] for all $f_1,f_2\in\mathcal{S}$.  Find the number of elements $F\in\mathcal{T}$ such that if $f\in\mathcal{S}$ satisfies $f(1)=2$ and $f(2)=1$, then $(F(f))(1)=2$ and $(F(f))(2)=1$.

Let's think step by step and output the final answer within \boxed{}.
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