Class Preserving Automorphisms of Some Nilpotent Groups of Class 2

An automorphism $$\alpha$$ of a group $$G$$ is called a class preserving automorphism if $$\alpha(g)$$ and $$g$$ are conjugate in $$G$$ for each $$g \in G$$ . We prove that each class preserving automorphism of the following nilpotent groups of class 2 is inner: (i) The direct product of a generalized extraspecial $$\mathbb{Z}$$ -group and a free abelian group with finite rank. (ii) An extension of $$\mathbb{Q}$$ by a direct sum of finitely many copies of $$\mathbb{Q}$$ , where $$\mathbb{Q}$$ is the additive group of rational numbers. (iii) An infinite Černikov $$p$$ -group which is not abelian but each proper quotient group is abelian.