Employer overpaid me after I ended employment. They have asked for that money back, but use a 3rd party payroll company. Should I contact payroll company to see if previous employer has already been reimbursed for payroll company's mistake, before I start

Company overpaid me after I quit, without me realizing. They have asked for money back, I have agreed to payment plan, because money was spent. However, they use a 3rd party payroll company, that they are claiming is at fault for this. Before I start repaying previous employer, would it be in my best interest to inquire with payroll company about whether or not previous employer has some sort of reimbursement contract with them? I will pay, but if my previous employer has already been reimbursed for this payroll company's mistake, then it's not my previous employer I owe. Let me say that I am willing to pay everything back, and I know that money is not mine to keep, but since it's a new calender year and this occurred at the end of the last, it is going to affect my taxes. What should I do?