= Solution
Choose $\widehat\zeta\in H^2(\widehat\Gamma,M)$ with $\iota^*(\widehat\zeta)=\zeta$, and let
$$
1\longrightarrow M\longrightarrow E\longrightarrow\widehat\Gamma\longrightarrow1
$$
be its extension. By the given fact, $E$ is a <profinite group>, hence is <residually finite>. The extension $E'_\zeta$ over $\Gamma$ is the pullback of $E$ along the injective map $\iota$. Part 5(a)(iii) embeds $E'_\zeta$ into $E$. Since every <subgroup of a residually finite group> is residually finite, so is $E'_\zeta$.
Back to article page
