Here's a question I should know the answer to but don't:
Suppose $1\to F \to G \to G/F \to 1$ is a short exact sequence of abelian groups with $F$ finite and $G/F$ torsion-free. Must the sequence split?
This is not true if you merely assume that $F$ is torsion. A counterexample is given by YCor here: https://mathoverflow.net/a/314536/20598.
Equivalently, suppose $G$ is a compact abelian group with finitely many components. Then does $G_0 \to G \to G/G_0$ split? This is not true without assuming there are finitely many components, as YCor's example shows, and it's also not true for nonabelian groups, as Max's answer here shows: https://math.stackexchange.com/a/954539/23805 (though I think it's true whenever $G/G_0$ is cyclic).