Is it permissible or impermissible for the bank to purchase a car in its entirety from the car dealer, and then sell it to the buyer in installments with an added interest?

This transaction is prohibited because it is an interest-based loan (riba). What the bank does is pay the price on behalf of the buyer and then collect it from him with an increase, and this is exactly the interest that Allah has forbidden. The permissible alternative is for the bank to buy the commodity, take possession and guarantee of it, and then sell it to the customer with the agreed-upon profit. This is the sale that Allah has permitted. The Prophet, peace and blessings be upon him, guided to the permissible way out of interest. Therefore, it should not be taken lightly with the argument that the outcome is the same, for the Sharia has objectives and wisdoms that may be hidden from some.