Let $G$ be a finite group acting trivially on an abelian group $A$. Let $\alpha$, $\beta \in Z^2(G,A)$ such that $\alpha\beta^{-1}\in B^2(G,A)$ . Let $S$ be a normal subgroup of $G$ and take $\alpha'=\alpha|S\times S$ and $\beta'=\beta|S\times S$. For $g\in G$, we define the $2$-cocycle $\varepsilon$ by $\varepsilon(s_1,s_2) = \alpha'(s_1,s_2)\beta'(g^{-1}s_1g,g^{-1}s_2g)^{-1}$ for all $s_1$, $s_2 \in S$. I try to prove that $\varepsilon \in B^2(S,A)$. In fact I did some computation by using the $2$-cocycle condition, but to get the required result we must have $\alpha(s,g)=\beta(s,g)$ for all $s\in S$. So it remains to show the last equation, but I don't know why it is true.
Any help would be appreciated so much. Thank you all.