Order of an automorphism

Definition
Let $$\varphi$$ be an defining ingredient::automorphism of a group $$G$$. The order of $$\varphi$$ is defined as its order as an element of the automorphism group $$\operatorname{Aut}(G)$$. Equivalently, it is the order of the cyclic subgroup $$\langle \varphi\rangle$$ inside $$\operatorname{Aut}(G)$$.