Is it permissible to trade in electrical appliances if the buyer requests a sum of money to purchase an item at his own discretion, and then that is considered as if the seller sold it to him in installments?

This transaction is usury (riba) because it is a loan with interest disguised by a symbolic, fictitious deferred sale. Therefore, it must be avoided, and one must repent from it. It is possible to conduct a transaction through a legitimate Murabaha sale, by selling him the item he desires after you have purchased it and it has come under your responsibility. There is no harm in you appointing him as an agent for the purchase. After that is completed, and it comes under your responsibility, you then sell it to him in a new sale.