The automorphism group of finite $2$-groups associated to the Macdonald group

We consider the Macdonald group $⟨x,y\mathrm{\mid }{x}^{[x,y]}=x^{1+2^m\mathrm{\ell }},y^{[y,x]}=y^{1+2^m\mathrm{\ell }}⟩$ and its Sylow 2-subgroup $J=⟨x,y\mathrm{\mid }{x}^{[x,y]}=x^{1+2^m\mathrm{\ell }},y^{[y,x]}=y^{1+2^m\mathrm{\ell }},x^{2^{3m-1}}=y^{2^{3m-1}}=1⟩$, where $m\ge 1$ and $\mathrm{\ell }$ is odd. Then $J$ has order $2^{7m-3}$, and nilpotency class 5 if $m>1$ and 3 if $m=1$. We determine the automorphism group of the 2-groups $J$, $H=J\mathrm{/}Z(J)$ and $K=H\mathrm{/}Z(H)$, where $\mathrm{\mid }H\mathrm{\mid }=2^{6m-3}$ and $\mathrm{\mid }K\mathrm{\mid }=2^{5m-3}$. Explicit multiplication, power, and commutator formulas for $J$, $H$, and $K$ are given, and used in the calculation of $\mathrm{A}\mathrm{u}\mathrm{t}(J)$, $\mathrm{A}\mathrm{u}\mathrm{t}(H)$, and $\mathrm{A}\mathrm{u}\mathrm{t}(K)$.