Is it permissible for the bank to pay for a portion of the car's price in installments, and then pay the remaining portion in installments after a period, knowing that the bank purchased the car directly from the dealership, and the buyer signed an authorization for the transfer of the amount from the bank to the dealership?

Purchasing a car through a bank takes two forms: The first is when the bank's role is purely financial, and this is a usurious loan that is forbidden. The second is when the bank buys the car and takes possession of it, then sells it to the customer. This is permissible under certain conditions: that the bank does not stipulate a late payment penalty, that the customer does not sign the purchase contract before the bank owns the car, that no down payment is required before ownership, that the contract effectively transfers ownership to the buyer immediately upon signing, and that comprehensive insurance is not required if it is commercial. It appears from the question that the transaction is a usurious loan, as the bank is a financier, not a seller. What confirms that it is a usurious loan is that when the purchase of the first car was canceled, the bank did not revoke the sale and initiate a new sale for the other car after purchasing it; rather, it increased the loan amount. If it is possible to annul this usurious contract, then it must be done, based on the hadith of the Prophet, peace and blessings be upon him, who commanded the annulment of a sale that involves usury. If it is impossible to annul the transaction, then there is no blame in proceeding with it, especially given your ignorance of the situation.