Automorphisms of finite groups

We study the classification of those finite groups G having a non-inner class preserving. automorphism. Criteria for these automorphisms to be inner are established. Let G be a nilpotent-by-nilpotent group and S is an element of Syl(2)(G). If S is abelian, generalized quaternion or S is dihedral, and in this case G is also metabelian, then Out(c)(G) = 1. If S is generalized quaternion, L(S) subset of L(G) and S-4 is not a homomorphic image of G, then Out(c)(G) = 1. As a consequence, it follows that the normalizer problem of group rings has a positive answer for these groups.