“I made an error in judgment in sharing a post made by a colleague on my private Facebook page which was a summer job notice, and indicating there might be a political preference in the selection process,” she said.

Ms. Oxender did not respond to questions on Wednesday about the identity of that person or what, if any, position he or she occupies in state or local government or the Republican Party.

The Department of Transportation denied that beliefs play any role in how young people are hired for summer jobs, a process handled at the local level in each of its 12 districts across the state.

When asked if the post written or shared by Ms. Oxender reflected the department’s hiring process, a spokesman, Matt Bruning, replied: “Oh God, no. No.”

Mr. Bruning added that a local politician like Ms. Oxender would not be involved in evaluating candidates or extending a job offer. The department looks at applicants’ qualifications, not their voting records, he said.

“I have no idea why she felt she needed to say something about political parties’ being part of the equation because that is not something that we do,” he said. “To write that there is preferential treatment given to a member of a political party is insane.”

Mr. Bertram, the city legal director and a former Republican City Council president, denied that political cronyism played any role in how government jobs were given out in Marietta. He said doing so would not be “kosher.”

He called Ms. Oxender’s Facebook post “an off-the-cuff statement that was inappropriate that has been blown out of proportion,” but said he had trouble understanding why she would have written or shared it at all.