In this short survey article, we try to list maximum number of known results
on class preserving automorphisms of finite $p$-groups. We conclude the survey
with some interesting (at least to me) open problems on this topic.
We study finite capable $p$-groups $G$ of nilpotency class 2 such that the
commutator subgroup $\gamma_2(G)$ of $G$ is cyclic and the center of $G$ is
contained in the Frattini subgroup of $G$.
Let $1 \to N \to G \to H \to 1$ be an abelian extension. The purpose of this
paper is to study the problem of extending automorphisms of $N$ and lifting
automorphisms of $H$ to certain automorphisms of $G$.