Given a group $G$ and a short exact sequence
$$1 \longrightarrow G \longrightarrow E \longrightarrow \mathbb{Z}/2\mathbb{Z} \longrightarrow 1$$
does the extension always split? That is, is it always true that $E = G \rtimes \mathbb{Z}/2\mathbb Z$?
2026-03-25 06:08:58.1774418938
Does every $G$-by-$C_2$ extension split?
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No, e.g. $E$ could be $\mathbb{Z}/4\mathbb{Z}$.