Modules over group rings of groups with restrictions on the system of all proper subgroups

We consider the class M of R-modules where R is an associative ring. Let A be a module over a group ring RG, G be a group and let L(G) be the set of all proper subgroups of G. We suppose that if H ∈ L(G) then A/CA(H) belongs to M. We study an RG-module A such that G ≠ G , CG(A) = 1. We consider the cases: 1) M is the class of all artinian R-modules, R is either the ring of integers or the ring of p-adic integers; 2) M is the class of all finite R-modules, R is an associative ring; 3) M is the class of all finite R-modules, R= F is a finite field.