Fix a prime $p$, and let $M$ be the unique group of order $p^3$ and exponent $p$. Let us denote by $E_n$ the elementary abelian group of rank $n$.

Is it true that $\operatorname{Aut}(M \times E_n)$ and $\operatorname{GL}(n+3,p)$ have isomorphic Sylow $p$-subgroups?

For $n=0$ they are isomorphic; and If I'm not mistaken they have the same order for all $n$.

Thanks in Advance.

Fix a prime $p$, and let $M$ be the unique nonabelian group of order $p^3$ and exponent $p$. Let us denote by $E_n$ the elementary abelian group of rank $n$.

Is it true that $\operatorname{Aut}(M \times E_n)$ and $\operatorname{GL}(n+3,p)$ have isomorphic Sylow $p$-subgroups?

For $n=0$ they are isomorphic; and If I'm not mistaken they have the same order for all $n$.

Thanks in Advance.