Past exam of the mathematics course of the University of Cambridge / 2018 / ib / Paper 4 / 2G / b / Solution

Let $D_8=⟨r,s:r^4=s^2=1,srs=r^{-1}⟩$, the symmetry group of a square. The mappreserves the relations and is an automorphism. Conjugation by a rotation sends $s$ to $r^{2j}s$, while conjugation by a reflection also inverts $r$. Therefore no conjugation fixes $r$ and sends $s$ to $rs$. This is an outer automorphism of the nonabelian group $D_8$.