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 \(\beta \ne 1\) , then \(G(\beta )\) is of class 3 and \(\langle B,C\rangle \) is a finite metacyclic group of order \(|\beta -1|^3\) , 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 \(\beta \ne 1\) , then \(|\textrm{Aut}(G(\beta ))|=2(\beta -1)^4\) and we exhibit an imbedding \(\textrm{Aut}(G(\beta ))\hookrightarrow {\textrm{GL}}_4({\mathbb {Z}}/(\beta -1){\mathbb {Z}})\) , 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 [3], 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 \(\textrm{Aut}(G(\beta ))\rightarrow \textrm{Aut}(L(\beta ))\) is injective and \(\textrm{Aut}(L(\beta ))\) is an extension of the Heisenberg group over \({\mathbb {Z}}/(\beta -1){\mathbb {Z}}\) direct product \(C_{\beta -1}\) , by the holomorph of \(C_{\beta -1}\) .
