Let $p$ be a Merssene prime, i.e. $p=2^a-1$, where $a$ is a prime.

Let $R$ be a 2-group of order $2(p+1)=2^{a+1}$. Also we know that $|Z(R)|=2$ and $R/Z(R)$ is abelian.
Can we conclude that $R$ has no automorhism of order $p$?

I know that there is a theorem that says that if $p$  is a prime and $G$  is a $p$ -group with $|G|=p^n$, $|Aut(G)|$  divides $\prod_{k=0}^{n-1} (p^n −p^k)$. But this is not enough for getting this result.

Thanks for your helps