[Math] Relations between the cohomology of discrete groups and of profinite groups

Question

Let $G$ be a discrete group and $K$ be the profinite completion of $G$. Let $C_K$ denote the category of contionuous $K$-modules and ${C_K}'$ denotes category of finite continuous $K$-modules. Now for $A \in {C_K}'$, I am trying to prove the restriction map
\begin{equation} H^q(K,A) \rightarrow H^q(G,A) \end{equation} is surjective for all $q \leq n$ if for all $x \in H^q(G,A)$, $1\leq q\leq n$, there exist an $A' \in C_K$ (not necessarily finite) such that $x$ maps to $0$ in $H^q(G,A')$.
This is an exercise in Serre's Galois Cohomology. I am able to prove the statement is valid for $n=1$ without assuming the existence of $A'$. For arbitrary $n$, my idea is using the long exact sequence of cohomology groups obtained from \begin{equation} 0 \rightarrow A \rightarrow A' \rightarrow A'/A \rightarrow 0 \end{equation} and doing induction on $n$. However since $A'$ is not necessarily finite this argument does not work. Is there any way of replacing $A'\in C_K$ with some abelian group in ${C_K}'$ ? or am I missing something ? Thanks,
Ozlem

Answer 1

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 ?