$\DeclareMathOperator\res{res}$ Let $G$ be a profinite group. Let $A$ be a finite $G$-group (a finite group on which $G$ acts continuously), not necessarily abelian, with center $Z=Z(A)$. We say that a second cohomology class $\xi\in H^2(G, Z)$ is neutral if $$\xi\in\, {\rm im}\big[\delta\colon H^1(G,A/Z)\to H^2(G,Z)\big]. $$ If $A$ is abelian, then $\xi$ is neutral if and only if $\xi=0$.
Let $G_1,\dots,G_r\subset G$ be open subgroups of indices $n_1,\dots,n_r$, respectively, where $\gcd(n_1,\dots,n_r)=1$. We consider the restriction maps $$\res_i\colon H^2(G, Z)\to H^2(G_i,Z).$$
Question. For $\xi\in H^2(G, Z)$, assume that the restrictions $\res_i(\xi)\in H^2(G_i,Z)$ are neutral for all $i=1,\dots, r$, where $\gcd(n_1,\dots,n_r)=1$. Does it follow that $\xi$ is neutral?
If $A$ is abelian, then the answer is Yes: using the corestriction maps, from $\res_i(\xi)=0$ we deduce that $n_i\xi=0$ for each $i$, whence $\xi=0$ because $\gcd(n_1,\dots,n_r)=1$. In the general case, I would expect the answer No.
