  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.
