What is the ruling on a bank purchasing a house, taking possession of it, and mortgaging it, then selling it to the client in installments, and transferring its ownership to him after the full amount has been paid?

The scenario mentioned in the question is permissible, because Murabaha (cost-plus financing) is permissible with its Sharia-compliant conditions. The amount taken initially is a pledge (down payment) through which the bank ensures the continuation of the transaction. Some scholars have permitted taking a pledge before the transaction is concluded. Furthermore, the house remaining in the bank's name is a mortgage on its part until the debt is repaid, and there is no harm in mortgaging the purchased item itself.