Answer by ozheidi for Relations between the cohomology of discrete groups and of profinite groups

So I think I need to find a finite sub-$K$-module $A_{\alpha}$ of $A'$ containing $A$ such that $x \in H^q(G,A)$ maps to $0$ in $H^q(G, A_{\alpha})$.

Here is the idea : Writing $ A'\cong \varinjlim_{\beta} A_{\beta}$ where the limit is taken over all finitely generated sub-$K$-modules, we have a map \begin{equation} \phi :\varinjlim_{\beta} H^q(G,A_{\beta}) \rightarrow H^q(G,\varinjlim_{\beta}A_{\beta}) \end{equation}

Now if this map is injective, I can conclude that $x$ maps to $0$ on one of these $H^q(G,A_{\alpha})$'s and maybe taking the torsion part of $A_{\alpha}$ I may be able to find such a finite sub-$K$-module of $A'$ that kills the cohomology class $x$.

I know $\phi$ is not always an isomorphism but is it possible $\phi$ to be injective ?