The automorphism group of finite \(2\)-groups associated to the Macdonald group (Q6593811)

In this paper, the authors continue the intensive study of the group \(G=G(1+2^{m}\ell)=\langle x, y \mid x^{[x,y]}=x^{1+2^{m}\ell}, y^{[y,x]}=y^{1+2^{m}\ell} \rangle\) (\(\ell\) odd), a special case of the groups introduced by \textit{I. D. Macdonald} in [Can. J. Math. 14, 602--613 (1962; Zbl 0109.01502)], as early examples of finite deficiency zero nontrivial groups.\N\NLet \(J\) be the Sylow \(2\)-subgroup of \(G\), \(H=J/Z(J)\) and \(K=J/Z_{2}(J)\). The structure of \(J\) is analyzed by the authors in [J. Group Theory 27, No. 3, 549--594 (2024; Zbl 1537.20049)] and here they devote themselves to describing \(\mathrm{Aut}(J)\), \(\mathrm{Aut}(H)\) and \(\mathrm{Aut}(K)\).