Is it permissible for a person to buy goods for another person with his own money, then sell them to him in installments with an increase in price, knowing that he does not own a shop and is not a merchant?

There is no objection to purchasing an item to sell it to someone who requested it, whether the price is paid in installments or not, or whether it is more or less than what you paid for it. A condition for the validity of this transaction is that the contract for the item must not be concluded before the seller purchases it, based on the Prophet's (peace be upon him) saying: "Do not sell what you do not possess." It is not a condition for the validity of the sale to own a shop or for the person to be a merchant.