The automorphism group of certain polycyclic groups

For $\beta\in{\mathbb Z}$, let $G(\beta)=\langle A,B\,|\, A^{[A,B]}=A,\, B^{[B,A]}=B^\beta\rangle$ be the infinite Macdonald group, and set $C=[A,B]$. Then $G(\beta)$ is a nilpotent polycyclic group of the form $\langle A\rangle\ltimes\langle B,C\rangle$, where $A$ has infinite order. If $\beta\neq 1$, then $G(\beta)$ is of class 3 and $\langle B,C\rangle$ is a finite metacyclic group of order $|\beta-1|^3$, which is an extension of $C_{(\beta-1)^2}$ by $C_{|\beta-1|}$, split except when $v_2(\beta-1)=1$, while $G(1)$ is the integral Heisenberg group, of class 2 and $\langle B,C\rangle\cong{\mathbb Z}^2$. We give a full description of the automorphism group of $G(\beta)$. If $\beta\neq 1$, then $|\mathrm{Aut}(G(\beta))|=2(\beta-1)^4$ and we exhibit an imbedding $\mathrm{Aut}(G(\beta))\hookrightarrow {\mathrm GL}_4({\mathbb Z}/(\beta-1){\mathbb Z})$, but for the case $\beta\in\{-1,3\}$ when 5 is required instead of 4. When $\beta$ is even the automorphism group of $\langle B,C\rangle$ can be obtained from the work of Bidwell and Curran \cite{BC}, and we indicate which of their automorphisms extend to an automorphism of $G(\beta)$. In general, we give necessary and sufficient conditions for $G(\beta)$ to be isomorphic to $G(\gamma)$. When $\gcd(\beta-1,6)=1$, we determine the automorphism group of $L(\beta)=G(\beta)/\langle A^{\beta-1}\rangle$, which is a relative holomorph of $\langle B,C\rangle$, and $\langle A^{\beta-1}\rangle$ is a characteristic subgroup of $G(\beta)$. The map $\mathrm{Aut}(G(\beta))\to \mathrm{Aut}(L(\beta))$ is injective and $\mathrm{Aut}(L(\beta))$ is an extension of the Heisenberg group over ${\mathbb Z}/(\beta-1){\mathbb Z}$ direct product $C_{\beta-1}$, by the holomorph of $C_{\beta-1}$.