I'm confused on the facts, but look up Repayment under Claim of Right.

Sounds like an elaborate phishing scam.  But then, so does a lot coming out of Washington these days.  If she plays along, she's likely to pay tax on what they say they have to send, and not be able to deduct what they want her to send back.  I'd ask a local member of Congress for help.  

It is quite confusing, but she worked for US game and fish in 2019 and then transferred to US forest service.  After the transfer, game and fish contacted her stating that she was paid for a pay period for which she had already left their employ.  She was able to prove when she left employment, so because a bill had been generated in their system they decided to pay her the amount she owed, and she had to give it back upon deposit of the check.....now, she has received a W2 for 2024 for the amount paid, including social security and medicare withholding. 

I'm still confused.

In 2019 she was paid the proper amount. Resolved after some back and forth.

In 2024 she was paid the "disputed" amount again?

Then in 2024 she had to repay the 2024 payment?

But she got a 2024 W-2 for the amount she received in 2024 that she paid back?

Or something different?

You have articulated it as I understand it. 

The Game and Fish apparently went through an audit sometime after she left employment, and this discrepancy in pay was discovered. So they generated a bill to her for the "overpayment of wages", which she dealt with.  Since, according to them, they could not just void/delete the invoice, they decided to pay her for the amount of the invoice and have her make repayment to close out the invoice.  Now she has received a W2 for the amount of the "repayment".

Then my original answer "look up Repayment under Claim of Right." still seems to be good.

Thank you for your help.

I will look it up.

You're welcome.

There is no federal "game and fish," those are state agencies.  Perhaps you meant Fish & Wildlife Service.  Or perhaps the scammer was mistaken.