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Let $G$ be a group (finite, say) with center $Z$. The automorphism group $\text{Aut}(G)$ sits in a short exact sequence $$ 1 \to G/Z \to \text{Aut}(G) \to \text{Out}(G) \to 1. $$ So when $Z\neq 1$, as ...

$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\End{End}\DeclareMathOperator\GL{GL}\DeclareMathOperator\Z{Z}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\Span{Span}\DeclareMathOperator\ker{ker}\...