Is it permissible to purchase a car in installments from a financier who deposits the amount into the buyer's account, then the buyer proceeds to purchase the car and sell it to the financier, who then sells it back to the buyer with a legitimate contract that includes a profit margin? Or is this transaction questionable due to the bank's involvement as an intermediary? And what should be done in case it is impermissible, given the potential financial loss?

This transaction is impermissible due to the involvement of the usurious bank in paying the money, whether directly to you or to the financier. The bank lends with interest, and your consent to this arrangement constitutes aiding the financier in sin and aggression by borrowing from usury to finance you. Allah has forbidden this, saying: "And cooperate in righteousness and piety, but do not cooperate in sin and aggression. And fear Allah; indeed, Allah is severe in penalty." (Al-Ma'idah: 2).

Selling the car is not necessary unless its sale becomes the only way to rid oneself of a religious prohibition. For example, if its price is a debt owed by you to the bank with usurious interest, and you cannot eliminate or reduce it except by selling the car and settling the debt at once, acting upon His (Allah's) saying: "O you who have believed, fear Allah and give up what remains [due to you] of usury, if you should be believers." (Al-Baqarah: 278), and His (Allah's) saying: "So fear Allah as much as you are able" (At-Taghabun: 16).