Let G be a group, and consider an automorphism \(\alpha \) of G. We call \(\alpha \) a commuting automorphism if it satisfies the condition \(\alpha (x)x= x \alpha (x)\) for all \(x \in G\) . The collection of all commuting automorphisms of G is denoted by \(\mathcal {A}(G)\) . The set \(\mathcal {A}(G)\) does not necessarily form a subgroup of the automorphism group of G. If \(\mathcal {A}(G)\) forms a subgroup of the automorphism group, then we say G is an \(\mathcal {A}\) -group. Let G be a finite non-abelian p-group given by a central extension of the form: \(1 \rightarrow \mathbb {Z}_{p^m} \rightarrow G \rightarrow \mathbb {Z}_{p} \times \cdots \times \mathbb {Z}_{p} \rightarrow 1\) , and its commutator subgroup has order p. Such a group G is known to be a generalized central product, \(G = E \circ A\) , where E is a generalized extraspecial p-group and A is an abelian group such that \(A = Z(G)\) and \(E \cap A = Z(E)\) . Let \(|E| = p^{2n+m}\) , \(|Z(E)| = p^m\) and \(|A| = p^{m+l}\) , where \(n,m \ge 1\) and \(l\ge 0\) . In this paper, we show that if \(n >1\) , then G is a non- \(\mathcal {A}\) -group if one of the following conditions holds: (i) \(p>2\) , (ii) \(p=2\) and \(m >1\) , (iii) \(p=2\) , \(m=1\) , and \(E\cong E_1\) , where \(E_1\) is a central product of n copies of dihedral groups of order 8, (iv) \(p=2\) , \(m=1\) , \(n>2\) , and \(E \cong E_2\) , where \(E_2\) is a central product of \(n-1\) copies of dihedral groups of order 8 and a quaternion group. Additionally, if \(n=1\) , then G is an \(\mathcal {A}\) -group.
