Cohomological variety in case that Sylow subgroup is elementary abelian

Question

Let $G$ be a finite group, $p$ a prime number, and $k$ an algebraically closed field of characteristic $p$. Then we can consider the cohomological variety of $G$, namely the maximal spectrum $V_G$ of the graded algebra $H^\bullet(G,k)=\operatorname{Ext}^\bullet_{k[G]}(k,k)$. By the Quillen stratification, this space has a stratification by quotients of the cohomological varieties of elementary abelian subgroups $E$ of $G$ (where an elementary abelian subgroup is one of the form $(\mathbb{Z}/p\mathbb{Z})^n$). To be precise, for an elementary abelian subgroup $E$, one has a map $V_E/W_E\to V_G$, where $W_E=N_G(E)/C_G(E)$. These maps are inseparable isogenies away from the locus cutting out smaller subgroups $E'$ of $E$, and together they give our stratification.

Now my $\textbf{question}$ is: if the Sylow $p$-subgroup $P$ of $G$ is itself elementary abelian, then do we obtain that $V_G$ is simply $V_P/W_P$?

Clearly this will be true on an dense open locus; further it holds in the case when $P$ is normal by the Lyndon-Hochschild-Serre spectral sequence. But I'm not clear about the general case.

Answer 1

Yes. There is a stronger result too, which predates Quillen's theorem: if $G$ is a finite group whose Sylow $p$-subgroup $P$ is abelian, then the restriction map $H^*(G;\mathbb{F}_p)\rightarrow H^*(P;\mathbb{F}_p)$ has image equal to the fixed points for the action of the normalizer $N_G(P)$ on $H^*(P;\mathbb{F}_p)$. Here I'm using $\mathbb{F}_p$ to denote the field of $p$ elements; the same theorem for other fields of characteristic $p$ follows. This theorem is due to Swan in "The $p$-period of a finite group", Ill. J. Math 4 (1960) 341-346. Or see Theorem II.6.8 of the book "Cohomology of Finite Groups" by Adem and Milgram.