Automorphism group of finite $p$-group when automorphism group of its quotient is $p$-group

Question

I was reading articles and books on automorphism group. It is always an interesting question to decide when the automorphism group is a $p$-group.

In this regard my question is

Let $\gamma(H)$ be the last non-trivial term of lower-central series of a finite $p$-group $H$. Suppose automorphism group of $G/\gamma(H)$ is a $p$-group. Then prove or disprove that the automorphism group of $H$ be a $p$-group

Sorry for not showing much effort from my side, I will appreciate any help.

Thanks in advance.