Given an elementary abelian $p$-group $E$, we can consider $E$ as a trivial $E$-module; my first question is :
How can one compute the rank of of the cohomology group $\operatorname{H}^n(E,E)$, $n \geq 2$?
I'm essentially interested to $\operatorname{H}^2(E,E)$.
Let $G$ be an extension of $E$ by $E$. How much it is difficult to classify all such groups $G$ having the property $\Omega_1(G)=G^p$? Is it true that every such a group is abelian (so homocyclic of exponent $p^2$) if $p$ is odd.
Thanks in advance.