Is it permissible to invest a sum of money with a relative who sells appliances in installments or for cash, such that I pay the price of the appliance in cash after the customer receives it, and the customer pays the installments to my relative, then the profits are divided between us? And what is the correct course of action if this is impermissible? And is it permissible for me to arrange installments for the customer without my relative's knowledge if I was the one who directed the customer to him?

If your relative sells the commodity through murabaha to a party who requests the purchase, meaning after he has taken possession and seized the commodity and it has entered his guarantee, then there is no harm in it. In this case, you may participate with him with capital on the basis of Mudarabah, such that you share the profit according to the agreement, and the loss is borne by the capital.

However, if the sale was completed and the customer received the device, and then you paid him its price in advance, then this money is not considered a Mudarabah for which you are entitled to profit. Rather, it is a guaranteed loan on which an increase is not permissible; otherwise, it would be Riba.