Upper bound on size of second cohomology group for groups of prime power order

Statement

Suppose $G$ and $A$ are both finite p-groups for some prime number $p$, with $A$ an abelian p-group. Suppose $G$ has order $p^n$ and $A$ has order $p^m$. Suppose $\varphi:G \to \operatorname{Aut}(A)$ is a group action. Then, we have the following upper bound on the size of the second cohomology group $\! H^2(G;A)$. The group $\! H^2(G;A)$ is itself a finite p-group and its prime-base logarithm of order is bounded as follows: $\! \log_p(|H^2(G;A)|) \le \frac{mn(n + 1)}{2}$

This result holds both for the case of trivial group action and nontrivial group action, i.e., both the case where $\varphi$ is a trivial map and the case where it is a nontrivial map.

Equality occurs in cases where both $G$ and $A$ are elementary abelian groups and the action is a trivial action (can it ever occur in other cases too?).

Examples

Examples with trivial group action

We consider here some examples: $p$ $G$ $n$ (so $|G| = p^n$) $A$ $m$ (so $A$ has order $p^m$) $H^2(G;A)$ $\log_p|H^2(G;A)|$ $mn(n+1)/2$ Information on second cohomology
any group of prime order 1 group of prime order 1 group of prime order 1 1 second cohomology group for trivial group action of group of prime order on group of prime order
any cyclic group of prime-square order 2 group of prime order 1 group of prime order 1 3 second cohomology group for trivial group action of cyclic group of prime-square order on group of prime order
any elementary abelian group of prime-square order 2 group of prime order 1 elementary abelian group of prime-cube order 3 3 second cohomology group for trivial group action of elementary abelian group of prime-square order on group of prime order

Examples with nontrivial group action

We consider here some examples: $p$ $G$ $n$ (so $|G| = p^n$) $A$ $m$ (so $A$ has order $p^m$) Action $H^2(G;A)$ $\log_p|H^2(G;A)|$ $mn(n+1)/2$ Information on second cohomology
2 cyclic group:Z2 1 Klein four-group 2 permutes the two direct factors trivial group 0 2 second cohomology group for nontrivial group action of Z2 on V4
2 cyclic group:Z2 1 cyclic group:Z4 2 inverse map action cyclic group:Z2 1 2 second cohomology group for nontrivial group action of Z2 on Z4