Is it permissible for the husband to pay 500 EGP in cash on behalf of his wife to the merchant, and then collect monthly installments of 600 EGP from the wife's relative, thereby earning 100 EGP without the knowledge of either the merchant or the wife's relative?

This transaction occurs in two forms:

The first is that you purchase the clothes and take possession of them, then sell them to your wife's relative in installments. This is permissible, provided that the relative knows you are the real seller, in order to ensure that each party can hold accountable the one who sold to them.

The second form is that you do not buy the clothes but rather pay the money owed by the buyer in order to retrieve more than that amount. This is prohibited usury (riba).