The automorphism group of finite $p$-groups associated to the Macdonald group

Abstract

Given positive integers $p$ and $m$, where $p$ is assumed to be an odd prime, we determine the automorphism groups of $p$-groups $J$, $H$, and $K$ of orders $p^{7m}$, $p^{6m}$, and $p^{5m}$, and nilpotency classes 5, 4, and 3, respectively, all of which arise naturally from the Macdonald group $⟨x,y|x^{[x,y]}=x^{1+p^m\ell },y^{[y,x]}=y^{1+p^m\ell }⟩$, where $\ell \in \mathbf{Z}$ is relatively prime to $p$.