To what extent is it permissible for the bank to buy a car for me and then sell it to me by installments as part of the car loan program?

The ruling on this transaction depends on understanding its true nature. If the bank owns the car and takes possession of it, then sells it to you in installments without stipulating a late payment penalty, then this transaction is permissible. However, it is common for usurious banks not to purchase cars; instead, they lend the customer its price, and then the customer repays them with an increment in installments. This is prohibited usury.