Is it permissible, Islamically, to sell dollars at a price higher than the bank's rate, with the knowledge of the owner of the money and taking the difference, and then sending the amount in Egyptian pounds to their family at the bank's rate, knowing that these funds do not belong to the seller?

It is permissible for you to authorize your friend to exchange dollars for Egyptian pounds. It is also permissible, according to some scholars, for your fee to be the amount exceeding the bank's exchange rate, citing the statement of Ibn Abbas: "There is no harm in saying: 'Sell this garment, and whatever exceeds such-and-such is yours.'" Some have interpreted this permissibility as analogous to a muqarada (profit-sharing) contract, and this is what Ahmad and Ishaq answered.