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Let $P$ be a p-group. Denote by $Z(G)$ its center and $H^*(P,\mathbb{F}_p)$ its mod $p$ cohomology ring.

My general (vague) question is what can be said about the mod $p$ cohomology of the central extension $$1 \rightarrow Z(P) \rightarrow P \rightarrow P/Z(P)$$ and in particular when $Z(P)$ is an elementary abelian $p$-group?

More specifically, I am wondering the following:

Let $\varphi$ be an automorphism of $P$ which acts as identity on $H(Z(G),\mathbb{F}_p)$ and $H^* (P,\mathbb{F}_p)$. Does $\varphi$ then also acts as identity on $H^*(P/Z(P), \mathbb{F}_p)$?

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    $\begingroup$ In case when P is extraspecial, then the answer is yes, because corestriction is just a quotient map from the cohomology of elementary (which is a polynomial algebra generated by Bocksteins of H^1 = P/Z(P)*) to the cohomology of P. In general I suspect it's not true. $\endgroup$
    – Denis T
    Commented Aug 21 at 12:46
  • $\begingroup$ I imagine that you are aware of the Lyndon-Hochschild-Serre spectral sequence, with $E_2^{i,j}= H^i(P/Z)\otimes H^j(Z)$ and converging to a filtration of $H^*(P)$? As Denis T says, the answer to your question is probably `no' in general, but not many computations have been done. $\endgroup$
    – IJL
    Commented Aug 23 at 14:17

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