= Solution
Let $\widehat\zeta\in H^2(\widehat\Gamma,M)$ lie in the kernel of $\iota^*$, and represent it by a profinite extension
$$
1\longrightarrow M\longrightarrow E\xrightarrow{\pi}\widehat\Gamma\longrightarrow1.
$$
Its pullback to $\Gamma$ is split, so there is a homomorphism $s:\Gamma\to E$ satisfying $\pi s=\iota$. By part b, $s$ extends uniquely to a continuous homomorphism $\widehat s:\widehat\Gamma\to E$. The continuous maps $\pi\widehat s$ and the identity of $\widehat\Gamma$ agree on the dense image of $\Gamma$, hence agree everywhere. Thus $\widehat s$ is a section of $\pi$, the extension splits, and $\widehat\zeta=0$. Therefore
$$
\boxed{\iota^*:H^2(\widehat\Gamma,M)\longrightarrow H^2(\Gamma,M)\text{ is injective}}.
$$
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