Nilpotency of a kernel of the Quillen map (Q1326737)

Let \(G\) be a finite group, \(p\) a prime and \(\mathcal A\) the set of all elementary abelian \(p\)-subgroups of \(G\). [In Ann. Math., II. Ser. 94, 549-602 (1971; Zbl 0247.57013)] \textit{D. Quillen} considered the natural map \(\phi: H^* (G, \mathbb{Z}/p) \to \bigoplus_{A \in {\mathcal A}} H^*(A; \mathbb{Z}/p)\) and showed that \(\text{Ker }\phi \subset \sqrt {0}\). In this paper, for any positive integer \(n\) the authors consider a certain central extension \(0 \to (\mathbb{Z}/2)^ n \to G \to (\mathbb{Z}/2)^ n \to 1\) and prove that there is an element \(t_ n \in H^ 1 (G, \mathbb{Z}/2)\) satisfying: \(t_ n \in \text{Ker }\phi\), \(t^ n_ n \neq 0\), \(t_ n^{n + 1} = 0\) and \(t^ n_ n \in \text{Ker }(H^* (G, \mathbb{Z}/2) \to \bigoplus_{H \in {\mathcal H}}H^*(H))\) where \(\mathcal H\) denotes the set of all proper subgroups of \(G\). They are using the Serre spectral sequence associated with the central extension. [In Proc. Am. Math. Soc. 118, 339-343 (1993; Zbl 0788.20029)] \textit{G. S. Avrunin} and \textit{J. F. Carlson} obtained independently part of the above theorem.