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Let $k$ be a number field and $$ 1\to A \to B \to C \to 1$$ be a central extension of finite groups over $\mathcal{O}_k$ (the ring of integers of $k$), with $B$ non-commutative. Consider the induced map (in étale cohomology over $\mathcal{O}_k$) $$ \partial_B : H^1(O_k,C) \to H^2(O_k,A) $$ (or, more generally, replacing $\mathcal{O}_k$ with $\mathcal{O}_{k,S}$, the ring of $S$-integers, for a finite set $S\subset \Omega_k$).

Is there an example of $A,B$ and $k$ such that $\partial_B$ is NOT surjective? Is it possible to describe any general conditions on $A,B$ and $k$ to ensure the non-surjectivity of $\partial_B$(e.g., $B$ is a $p$-group and $A$ a nontrivial cyclic central subgroup)?

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    $\begingroup$ If the extension is split then the boundary map is probably zero. So if the extension is a product of a split situation and a random non-commutative non-split situation then it's not in general going to be surjective, I shouldn't think. $\endgroup$
    – znt
    May 30, 2016 at 19:49

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