On commuting automorphisms and central automorphisms of finite \(2\)-groups of almost maximal class (Q6598021)

An automorphism \(\alpha\) of a group \(G\) is said to be central if \(g^{-1}g^{\alpha} \in Z(G)\) for every \(g \in G\) and commuting if \([g,g^{\alpha}]=1\) for every \(g \in G\). The set \(\mathrm{Aut}_{c}(G)\) of all central authomorphisms of \(G\) is a normal subgroup of \(\mathrm{Aut}(G)\). Conversely, the set \(\mathcal{A}(G)\) of all central automorphisms of \(G\) may not be a subgroup of \(\mathrm{Aut}(G)\), as \textit{M. Deaconescu} et al. have shown in [Arch. Math. 79, No. 6, 423--429 (2002; Zbl 1017.20028)]. If the set \(\mathcal{A}(G)\) is a subgroup of \(\mathrm{Aut}(G)\), then \(G\) is called \(\mathcal{A}\)-group.\N\NThe authors, in [J. Algebra Appl. 20, No. 4, Article ID 2150060, 17 p. (2021; Zbl 1517.20031)], proved that if \(G\) is a finite group of order \(2^{n}\) (\(n \geq 5\)) of almost maximal class, then \(\mathcal{A}(G)=\mathrm{Aut}_{c}(G)\), except five exceptions. In the paper under review, they determine the structure of \(\mathrm{Aut}_{c}(G)\) and \(\mathcal{A}(G)\) for such five exceptions. In particular, they characterize the upper central series of these groups and show that if \(G\) is a group of almost maximal class and order \(2^{n}\), \(n \geq 7\) in which \(2 \leq d(G') \leq 3\) (where \(d(G')\) denotes the minimum number of generators of \(G'\)), then \(\mathrm{Aut}(G)\) is an \(\mathcal{A}\)-group.