$\def\Z{\mathbb{Z}}$ Let $A$ be a finite abelian group and $G$ a finite group acting on $A$. Then the extension $0 \to A \to E \to G \to 1$ is splits if and only if the corresponding $2$-cocycle is trivial in $H^2(G,A)$.
Does there exist such a split extension $E$ such that $E$ can also be seen as a non split extension (for the same action of $G$ on $A$)? Or in other words, can we find two groups $G$ and $A$, and an action of $G$ on $A$ such that there exists $\alpha_1,\alpha_2 \in H^2(G,A)$ which corresponds to the same group $E$, and with $\alpha_1=0$?
With help from user gro-tsen, I can say the following:
In the answer here: https://math.stackexchange.com/questions/131881/group-extensions/131895#131895 one can find two extensions of $G=(\Z/p\Z)^2$ by $A=(\Z/p\Z)^2$, isomorphic as groups, and one splits and the other not. Unfortunately the action by $G$ on $A$ is different, so this does not quite answer my question.
In the answer here: Extensions isomorphic as groups but not congruent or pseudo-congruent one can find that there exist non pseudo congruent extensions (but isomorphic as groups), even for the trivial action, even for an abelian extension $E$. But in this example none of the two extensions split.
In fact if $E$ is abelian, it can't be both a split and a non split extension of $G$ by $A$. This is a special case of Can a module be an extension in two really different ways?
