Definition:Outer Automorphism

Definition
Let $G$ be a group.

Let $\phi: G \to G$ be an automorphism which is not an inner automorphism.

Then $\phi$ is an outer automorphism.

Also defined as
Some sources use the term outer automorphism to mean an element of $\dfrac {\Aut G} {\Inn G}$: the quotient group of the automorphism group by the group of inner automorphisms.

Because of such ambiguity in understanding what is meant, it is important to specify which definition is meant whenever the term is used.

On, the former definition, that is an element of $\Aut G \setminus \Inn G$, is meant.