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