Some results on C-automorphisms of finite nilpotent by nilpotent groups

In this paper, C-automorphisms of finite nilpotent by nilpotent groups are discussed. We first present a criteria for a C-automorphism of a nilpotent by nilpotent group to be an inner automorphism. As an application of our result, we prove that C-automorphisms of semidirect products of cyclic by nilpotent groups are inner automorphisms. We also prove that under some additional conditions C-automorphisms of extensions of nilpotent groups by cyclic 2-groups are inner automorphisms. As immediate consequences of our results, we obtain that the normalizer property holds for such groups.