Outer automorphism of a group ($\alpha \in \mathrm{Aut}(G)\setminus \mathrm{Inn}(G)$)

An outer group automorphism is not conjugation by any element of the group. Its nonidentity class belongs to $\mathrm{Out}(G)=\mathrm{Aut}(G)/\mathrm{Inn}(G)$. A permutation-group example is the exceptional automorphism of $S_6$ that swaps its natural and transitive copies of $S_5$; it sends a transposition to a product of three disjoint transpositions and hence cannot be inner.