In chapter 1, exercise $\S$2.1 of Neukirch's Cohomology of Number Fields page 24, there is the following situation: $G$ is a profinite group, $A$ is a discrete group (not necessarily abelian) with a continuous $G$-action, $\hat{G} = A \rtimes G$, and $\mathrm{SEC}(\hat{G}\to G)$ is defined as the $A$-conjugacy classes of homomorphic sections of the natural exact sequence $$1 \to A \to \hat{G} \xrightarrow{\pi} G \to 1.$$ The result in question is that if either $G$ or $A$ is finite, then there is a canonical bijection of pointed sets $H^1(G,A) \cong \mathrm{SEC}(\hat{G}\to G)$.
My questions are: (1) Why is it necessary for either $G$ or $A$ to be finite? (2) Would the result be true if $A$ were not discrete but profinite? (3) If $A$ were profinite, would the finiteness condition be necessary?