On inner automorphisms and certain central automorphisms of groups

Let G be a group, let M and N be two normal subgroups of G. We denote by AutNM (G), the set of all automorphisms of G which centralize G/M and N. In this paper we investigate the structure of a group G in which one of the Inn(G) = AutNM (G), AutNM(G) ≤ Inn(G) or Inn(G) ≤ AutNM(G) holds. We also discuss the problem: “what conditions on G is sufficient to ensure that G has a non-inner automorphism which centralizes G/M and N”.