The outer automorphism group of a group is the quotient of its automorphism group by its inner automorphism group: $$\mathrm{Out}(G)=\mathrm{Aut}(G)/\mathrm{Inn}(G).$$

There is some variance in terminology about ``an outer automorphism.'' Some authors define an outer automorphism as any automorphism $\phi:G\rightarrow G$ which is not an inner automorphism. In this way an outer automorphism still permutes the group $G$ . However, an equally common definition is to declare an outer automorphism as an element of $\mathrm{Out}(G)$ and consequently the elements are cosets of $\mathrm{Inn}(G)\phi$ , and not a map $\phi:G\rightarrow G$ . In this definition it is not generally possible to treat the element as a permutation of $G$ . In particular, the outer automorphism group of a general group $G$ does not act on the group $G$ in a natural way. An exception is when $G$ is abelian so that $\mathrm{Inn}(G)=1$ ; thus, the elements of $\mathrm{Out}(G)$ are canonically identified with those of $\mathrm{Aut}(G)$ so we can speak of the action by outer automorphisms.