What is the ruling on selling household appliances to customers by installment, knowing that the seller does not own the shop and first buys the appliances then sells them by installment for a profit? And is it permissible to sell them to someone who buys them with the intention of selling them to the same shop?

If you purchase a commodity desired by a customer, take possession of it, and then sell it to them on installment at a price higher than the cash price, there is no harm in that. The buyer selling the commodity to another store does not matter, as there is no relationship between them. Selling commodities after purchasing and possessing them is permissible in Islamic law, based on the Almighty's saying: ﴿And Allah has permitted trade﴾ [Al-Baqarah: 275], and His saying: ﴿O you who have believed, when you contract a debt for a specified term, write it down﴾ [Al-Baqarah: 282].

However, selling a commodity before purchasing and possessing it is not permissible, due to the Prophet's ﷺ prohibition of selling commodities where they are bought until merchants take possession of them to their abodes, and his ﷺ saying: "Do not sell what you do not possess."

The prohibited Bay' al-'Inah (transaction of 'Inah) is when one buys a commodity for a deferred price, then sells it back to the person who sold it to them for a lower price, and this is not the case here.