Is it permissible for a Facebook group owner to take a financial subscription in exchange for publishing marketing advertisements, and is it permissible to pay him this money?

It is permissible to pay fees for publishing an advertisement if it is free from prohibited elements, and it is permissible for the group to take fees for this service, because it is an ijarah (leasing/hiring contract), and ijarah is permissible if its Shariah conditions are met. Among these conditions is that the benefit derived must be permissible; thus, it is not permissible to advertise prohibited things like gambling or alcohol.