On commuting automorphisms of some groups

Let G be a group. If the set \({\mathcal {A}}(G)=\lbrace \alpha \in {\textit{Aut}}(G): x\alpha (x)=\alpha (x)x\; \textit{for all}\; x\in G\rbrace \) forms a subgroup of \({\textit{Aut}}(G)\), then G is called \({\mathcal {A}}\)-group. In this paper, we prove that a metacyclic group is an \({\mathcal {A}}\)-group. Also, we show that, for any positive integer n and any prime number p, there exists a finite \({\mathcal {A}}\) p-group of nilpotency class n. Since there exist finite non \({\mathcal {A}}\) p-groups with \(\vert G/G^{\prime }\vert = p^{4}\), we find suitable conditions implying that a finite p-group with \(\vert G/G^{\prime }\vert \le p^{3}\) is an \({\mathcal {A}}\)-group. Using these results, we show that there exists a finite \({\mathcal {A}}\) p-group G of order \(p^{n}\) for all \(n\ge 4\) such that \({\mathcal {A}}(G)\) is equal to the central automorphisms group of G. Finally, we use semidirect product and wreath product of groups to obtain suitable examples.