The cyclist who was fired after posting a picture of herself “flipping the bird” to President Donald Trump’s motorcade on social media does not have a case against her former employer based on her First Amendment right of free speech.

But, arguably, she could pursue gender discrimination charges because a male employee of the firm was not terminated after his offensive posting, although experts say she would likely have an uphill battle.

According to news reports, Ms. Juli Briskman, who oversaw social media for Herndon, Virginia-based Akima L.L.C., raised her finger in an obscene gesture to the Trump motorcade on Oct. 28 while it was on its way back from a golf course, then posted a picture of her gesture on her Facebook and Twitter accounts, which were later widely disseminated. She did not identify herself as an Akima employee.

Ms. Briskman said although she was fired for her action, another male employee of Akima, which is a government contractor, who had posted objectionable content about the Black Lives Matter movement was permitted to retract his comment and keep his job.

A spokeswoman for the American Civil Liberties Union in Washington said Ms. Briskman had approached the organization about representing her, but it has not taken her case. A blog by the organization on the case said she is not protected under the First Amendment, because it only prohibits the government from punishing speech.

It could not immediately be learned whether Ms. Briskman has approached others about pursuing litigation.

According to news reports, more than $50,000 has been raised for Ms. Briskman by a supporter who set up a GoFundMe camping.

Akima did not respond to a request for comment.

Experts say the situation is comparable to workers who were fired after their employers learned of their attendance at a white supremacists’ rally at Charlottesville, Virginia; the situation involving James Damore, the Google Inc. engineer, who was fired after criticizing the company’s diversity efforts; and National Football League players who have kneeled during the national anthem.

Experts point out that only a handful of states — which do not include Virginia, where Akima is based — would offer employees protection in a case such as this.

“They fired her legally,” said Eric Meyer, a partner with Dilworth Paxson L.L.P. in Philadelphia. “Virginia’s an at-will employment state, and as far as I know she was an at-will employee.”

“They could have any reason or no reason at all to get rid of her,” but in this case she violated the firm’s social media policy, “so that gilded the lily, so to speak,” said Mr. Meyer.

“Generally, an employer who finds out that an employee has done some political speech or political protest outside their working hours can take any action they want against that employee,” unless they are among the handful of states that have laws against that, said Demetri J. Economou, a Houston-based attorney with Kane Russell Coleman Logan P.C.

Employees use social media “at their peril,” said Richard B. Cohen, a partner with FisherBroyles L.L.P. in New York.

However, “You could make a claim in this case that she’s being discriminated against because she’s female,” which could rise to a gender discrimination claim, said J. William Manuel, a partner with Bradley Arant Boult Cummings L.L.P. in Jackson, Mississippi. “That’s probably going to be the basis for some sort of charge.”

The male executive was not fired, said Aaron Goldstein, a partner with Dorsey & Whitney L.L.P. in Seattle. “He was told to take down the information, and he did, and he went on his merry way,” he said.

“If I were that company’s lawyer, I’d be a little nervous about a potential gender discrimination claim because the female was treated so differently,” he said.

A likely defense, Mr. Goldstein added, is that Ms. Briskman could have damaged the company’s business as a government contractor. “If the company is squeaky clean and there’s no other hint of gender bias in the workplace, maybe they could prevail” on the basis the situation had nothing to do with gender, Mr. Goldstein said.

But, he added, “All you need is a few sexist jokes popping up on email to hurt that argument.”

Some experts say, though, that Ms. Briskman would face an uphill battle if she makes a gender discrimination claim.

“You have two different situations,” said Mr. Meyer, comparing the male executive with Ms. Briskman.

“He wasn’t featured on late-night television,” he said. “Her situation drew a heck of a lot more notoriety and attention for the company” and “put it on the map in terms of its role as a government contractor, so she potentially could have endangered the company’s role as a government contractor,” which the other executive did not.

Timm Schowalter, a shareholder with Sandberg Phoenix & von Gontard P.C. in St. Louis, said the incident may also raise the issue of whether the employer’s social media policy violates Section 7 of the National Labor Relations Act, which gives workers the right to engage in concerted activities. The Obama administration, he said, was active in enforcing this provision, although it is unclear whether the Trump administration will continue to do so.

But Joel Rice, a partner with Fisher Phillips L.L.P. in Chicago, said, “I don’t see her protected under the NLRA because she was not engaged in conduct that pertained to the workplace or her rights in the workplace.”

Mr. Schowalter also said if the company has employment practices liability insurance and there is litigation filed, it is likely the case would be settled, assuming the plaintiff’s demands are “reasonable,” rather than risk high legal fees.