= Group extension
{wiki}
A group extension of $G$ by a <group> $N$ is a <short exact sequence>
$$
1\longrightarrow N\xrightarrow{i}E\xrightarrow{\pi}G\longrightarrow1.
$$
Thus $i$ identifies $N$ with a <normal subgroup> of $E$, and $\pi$ identifies $E/N$ with $G$. The <group> $N$ need not be abelian. A lift of $g\in G$ acts on $N$ by <conjugation>; different lifts differ by an <inner automorphism>, so a general extension induces a <group homomorphism> $G\to\operatorname{Out}(N)$ to the <outer automorphism group>.
When $N=A$ is an <abelian group>, its <inner automorphisms> are trivial. The extension therefore specifies an actual <group action> on $A$, making it a $G$-<module>. Extensions with that fixed <module> structure are classified by <second group cohomology classifies group extensions>. For example, $C_4$ and $C_2\times C_2$ are extensions of $C_2$ by $C_2$ with trivial <group action>, but the first does not split and the second does. A <split group extension> is exactly a <semidirect product>.
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