Is it permissible to purchase a residence from a fund in installments, with a 20% down payment, and pay the remaining balance with a 7.75% increase over 30 years in variable installments subject to increase or decrease according to the fund's policy?

Selling by installments is permissible under conditions, including that the installment and the term are known. What you mentioned about increasing or decreasing installments leads to an unknown amount for the installment and the term. Therefore, it is not permissible for you to buy the house in this way.