On Automorphisms of Some Finite $p$-groups

We give a sufficient condition on a finite $p$-group $G$ of nilpotency class 2 so that $\Aut_c(G) = \Inn(G)$, where $\Aut_c(G)$ and $\Inn(G)$ denote the group of all class preserving automorphisms and inner automorphisms of $G$ respectively. Next we prove that if $G$ and $H$ are two isoclinic finite groups (in the sense of P. Hall), then $\Aut_c(G) \cong \Aut_c(H)$. Finally we study class preserving automorphisms of groups of order $p^5$ and prove that $\Aut_c(G) = \Inn(G)$ for all the groups of order $p^5$ except two isoclinism families.