Let $d>1$ be an odd integer. Given a simplicial set $X$ and $[\gamma]\in H^2(X,\mathbb{Z}/d)$, there exists a fibration $N\mathbb{Z}/d\to E\to X$, with $$E= X_\gamma:=N\mathbb{Z}/d\times_{\gamma} X.$$ Conversely, for any "central extension" $N\mathbb{Z}/d\to E\to X$, we have that $E\cong X_\gamma$, for some normalized cocycle $\gamma:X_2\to \mathbb{Z}/d$. I am interested in simplicial sets $X,E$ with a certain abelian-group-like structure and I would like to understand when $N\mathbb{Z}/d\to E\to X$ splits, assuming that we know explicitly $H^2(X,\mathbb{Z}/p)$.
For that purpose, a first step would be to understand the following particular case: let us assume that $d=p$ an odd prime and $X=(\mathbb{Z}/p)^n$. If a central extension $$\mathbb{Z}/p\to E\to (\mathbb{Z}/p)^n$$ satisfies that $E$ is abelian and $g^p=1$, for every $g\in E$, then $[\gamma]=0$. This is true by the classification of finite abelian groups, but this kind of classification theorem does not work more generally.
Is it possible to give a "cohomological" proof of the vanishing of $[\gamma]$? It means that we can assume $H^{\ast}((\mathbb{Z}/p)^n,\mathbb{F}_p)\cong \Lambda(x_1,\dots,x_n)\otimes \mathbb{F}_p(y_1,\dots.y_n)$ is known.
