Quotient group acts on abelian normal subgroup

Statement
Suppose $$G$$ is a group and $$N$$ is an Abelian normal subgroup. Then, the quotient group $$G/N$$ has a naturally induced action on $$N$$, i.e., there is a homomorphism:

$$G/N \to \operatorname{Aut}(N)$$

given as follows:

$$gN \mapsto c_g$$

where $$c_g$$ is conjugation by $$g$$ in $$G$$.

The action is faithful if and only if $$N$$ is also a self-centralizing subgroup.

We need $$N$$ to be normal for the conjugation action to define an automorphism of $$N$$, and we need $$N$$ to be Abelian for the map to be well-defined and independent of the choice of the coset representative.

Generalizations

 * Quotient group maps to outer automorphism group of normal subgroup
 * Quotient by centralizer acts on normal subgroup