Semi-extraspecial $p$-groups with automorphisms of large order

In this paper, we consider semi-extraspecial $p$-groups $G$ that have an automorphism of order $\mathrm{\mid }G:G^{\mathrm{\prime }}\mathrm{\mid }-1$. We prove that these groups are isomorphic to Sylow $p$-subgroups of ${\mathrm{S}\mathrm{U}}_3(p^{2a})$ for some integer $a$. If $p$ is odd, this is equivalent to saying that $G$ is isomorphic to a Sylow $p$-subgroup of ${\mathrm{S}\mathrm{L}}_3(p^a)$.