Is it permissible to increase the price of a house by agreement between the buyer and the appraiser, so that the 90% paid by the Islamic bank covers the full value of the house, knowing that the buyer will repay the full amount to the bank, and is it permissible for the buyer to take the increase if it occurs?

If the bank buys 90% of the house by installments, this is permissible under the concept of "diminishing partnership" (Sharikah Mutanaqisah), provided that the promise to sell is not binding, that a separate sales contract is concluded for each share at the market price at the time of sale, and that there is no guarantee for the shares at their initial value. As for colluding with the seller or appraiser to change the house price to the bank, this is Islamically forbidden due to the elements of lying, deception, and fraud involved therein.