<abstract><p>In this paper, we prove that all Coleman automorphisms of extension of an almost simple group by an abelian group or a simple group are inner. Using our methods we also show that the Coleman automorphisms of $ 2 $-power order of an odd order group by an almost simple group are inner. In particular, these groups have the normalizer property.</p></abstract>

The normalizer problem of integral group rings has been studied extensively in recent years due to its connection with the longstanding isomorphism problem of integral group rings. Class-preserving Coleman automorphisms of finite groups occur naturally in the study of the normalizer problem. Let G G be a finite group with a nilpotent subgroup N N . Suppose that G / N G\hspace{-0.0em}\text{/}\hspace{-0.0em}N acts faithfully on the center of each Sylow subgroup of N N . Then it is proved that every class-preserving Coleman automorphism of G G is an inner automorphism. In addition, if G G is the product of a cyclic normal subgroup and an abelian subgroup, then it is also proved that every class-preserving Coleman automorphism of G G is an inner automorphism. Other similar results are also obtained in this article. As direct consequence, the normalizer problem has a positive answer for such groups.

In this study, the approach is through the projective limits, and we obtain some characterizations of the Coleman automorphism groups of a split metabelian group and a generalized dicyclic group. An application of our results is a generalization of the result of Z. Li and Y. Li.