Is the husband sinful if he lends his sister a sum of money to buy a house, knowing that the source for repaying the debt is unemployment benefits that his sister obtains through fraudulent means and in violation of their conditions? What is the ruling on the money that the husband recovers from this debt? And must the sister return the money she obtained in this way if she wishes to repent?

Your husband's sister is not permitted to take unemployment benefits if she violates the conditions for their entitlement, because it is illicit money that must be returned. Your husband is permitted to lend her money, even if he knows that she will repay the loan from this illicit money, because your husband receives the money through a legitimate means, and the sin is on the one who earns the illicit money, and the loan becomes a binding debt. Previous fatwas have advised that a creditor has the right to reclaim his debt without inquiring about the source of the money, citing the hadith (definition) of Barirah and the principle that "a change in the cause of ownership takes the place of a change in the essence." Your husband should advise and remind her.