I have to wonder if they are in such a rush to get it back due to there being a law in your state legally entitling you to the money.

I was once over paid by about $1,500 in one paycheck, the company's payroll software somehow classified me working in two departments at once. I wanted to give it back, but the company's legal department stepped in and wouldn't allow it. General counsel came back and said by state law I was legally entitled to the money, and the company would've had to pay out double what I was overpaid in fines, penalties, and legal costs to the state labor commission to accept it back. General counsel also felt I'd likely have to pay income tax on the extra $1,500 whether I gave it back or not. So I didn't, and morally made up for it by just not counting my overtime hours for a few weeks.

While I believe you should attempt to give the money back, make sure your employer looks into the best situation for you to do that before you do. As bad as a $6000 hit is, a $12,000 one could be much worse if you're working for a small business.