A note on automorphisms of finite p-groups

Let $p$ be an odd prime and let $G$ be a finite $p$-group such that $xZ(G)\subseteq x^G$, for all $x \in G - Z(G)$ where $x^G$ denotes the conjugacy class of $x$ in $G$. Then $G$ has a non-inner automorphism of order $p$ leaving the Frattini subgroup $\Phi(G)$ element-wise fixed. DOI: 10.1017/S000497271200021