Is it permissible to charge a fee for allowing someone else to use my credit card for an interest-free installment purchase, provided they commit to monthly repayments?

It is not permissible to charge fees for settling another person's installments using a credit card; this is considered a loan with an increase, which is usury (riba). The solution is for you to purchase and own the commodity, then sell it to your friend with a permissible profit. It is necessary to first verify the permissibility of installment transactions using a credit card, and if they are impermissible, then the transaction is absolutely not allowed. To verify the permissibility of the transaction, one should refer to specialized scholars who have knowledge of the nature of these cards and transactions.