You're going to be in a tough spot, and depending on the amount, you might want to talk to a labor lawyer. I'll explain the problem first, and how to report it on your return second.
The problem is that you need to either take a special itemized deduction, or use a procedure called "IRC 1341 Claim of Right". The itemized deduction won't result in a full refund of the tax owed unless you already have more than $12,500 of other itemized deductions (or $25,000 if married). The claim of right procedure would definitely get your tax money back, but to use it, you must have had a reasonable belief at the time you received the money, that you had an unrestricted right to the money. And since you knew at the time you were being overpaid, you knew that you did not have an unrestricted right to the money. If you had repaid the money in 2019, the company should have issued a correct W-2 for 2019. If you file under IRC 1341, and if you were audited, you risk losing your case.
Your 2019 W-2 can't be corrected because the wages you received are what you actually received, even if they were wrong. If you made repayment during 2019 then the company should (and must) issue a corrected W-2 to reflect what you were actually paid. But if you haven't made the repayment yet, the 2019 W-2 can't be corrected.
If the overpayment amount is less than $3000 there is no tax relief available.
If the overpayment amount is more than $3000, you can either take a special itemized tax deduction for the amount of repayment, or you can take an IRC 1341 Claim of Right tax credit. To claim the credit, you figure out what your tax would have been in 2019 without the extra wages, and the difference is the amount of your credit. You have to figure the credit amount yourself, you can do this by preparing test tax returns (for 2019) with and without the additional wages. Then, you manually enter the credit on Schedule 3 line 6 of your 2020 tax return, which you can only do using the desktop version of Turbotax installed on your own computer (not online). That credit will come back to you as a refund of the amount of excess tax you paid on your 2019 tax return. If you take the special itemized deduction, it will be at the bottom of the Deductions and Credits page under the listing for "other uncommon deductions." The itemized deduction may save you something on your taxes depending on your other deductions, but probably not as much as the credit.
For state income tax, you will have to research your state, there is probably a similar Claim of Right procedure.
For social security and Medicare tax, the employer could issue a refund of that tax, and issue a corrected W-2 that would change box 3-6 (social security and medicare wages and tax paid) but not change box 1 (gross wages). If they don't, you can request a refund using IRS form 843. You will need a letter from your employer verifying that you were required to repay wages and also verifying that the company will not be making the refund themselves.
Remember, you file a 2019 return using the 2019 information which is correct as to what you actually were paid in 2019. If you make the repayment in 2020, any deduction or credit is claimed on your 2020 tax return.
However, as I indicated, you probably don't have a Claim of Right in your case since you knew you were being overpaid, so professional legal and tax advice may be in order. If you explain this to the company you might be able to negotiate a reduced repayment to account for the lost taxes.
