On group automorphisms which centralize the factor group by an abelian normal subgroup

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In his well-known paper [8], Shoda investigated the structure of the automorphism group of a finite abelian group. Each such automorphism group has a nilpotent normal subgroup the factor group by which is a direct product of general linear groups. In this paper we show that this kind of group occurs frequently within the automorphism groups of a wide class of groups: Given any group G, and, say, a finite abelian normal subgroup A of G, the group of automorphisms of G which centralize G/A has the structure roughly described above. The heart of the paper is Lemma 3.1, where a simple general relation between the group C,,,,(G/A)