Non-inner automorphisms of order $p$ in finite $p$-groups admitting cyclic center

Let $G$ be a finite non-abelian $p$-group admitting cyclic center and $p$ be an odd prime. In this paper, we prove that if $C_G(Z({\gamma }_3(G)G^p))\nleqq {\gamma }_3(G)G^p$, then $G$ has a non-inner automorphism of order $p$.