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 $p$?

Thanks for your helps