Outer automorphism group maps to automorphism group of center

Statement

Suppose $G$ is a group. Denote by $\operatorname {Out} (G)$ the outer automorphism group of $G$ and by $Z(G)$ the center of $G$. Denote by $\operatorname {Aut} (Z(G))$ the automorphism group of $Z(G)$. Then, there is a canonical homomorphism:

$\operatorname {Out} (G)\to \operatorname {Aut} (Z(G))$

defined as follows: for any outer automorphism class $[\sigma ]$ of $G$, pick a representative automorphism $\sigma$, and consider the restriction of $\sigma$ to $Z(G)$.

Proof that this is well defined: Since any inner automorphism restricts to the identity on $Z(G)$, the automorphism obtained by restriction is independent of the choice of representative.

Related facts

• Outer automorphism group maps to automorphism group of any characteristic central series