Automorphisms of \(p\)-groups given as cyclic-by-elementary Abelian central extensions (Q5945118)

Let \(G\) be a \(p\)-group such that for a normal subgroup \(H\) of order \(p\), \(G/H\) is elementary Abelian. The author uses the known structure of such a group, being a central product of an extra-special and an Abelian group, to elucidate the structure of \(\Aut G\). In a first step it is proved that \(\Aut G\cong\Aut_NG\rtimes\langle\theta\rangle\), where \(o(\theta)=p-1\) and \(\Aut_NG\) denotes the subgroup of \(\Aut G\) which fixes the Frattini-group of \(G\) elementwise. In a second step \(\Aut_NG\) is determined. This is achieved with the help of a homomorphism from \(\Aut_NG\) into the direct product of a symplectic group and a general linear group. As an application the wedge-decomposition of the classifying space of \(G\), located at \(p\), is determined.