Automorphisms of groups and converse of Schur's theorem

An automorphism of a group G is called an IA-automorphism if it induces the identity automorphism on the abelianized group G/G'. Let IA(G) denote the group of all IA-automorphisms of G. We classify all finitely generated nilpotent groups G of class 2 for which IA(G) is isomorphic to Inn(G). In particular, we classify all finite nilpotent groups of class 2 for which each IA-automorphism is inner. As consequences, we give surprisingly very easy proofs of converse of Schur's theorem and also prove some other related results.