Solution
= Solution
Let $D_8=\langle r,s:r^4=s^2=1, srs=r^{-1}\rangle$, the symmetry group of a square. The map
$$
r\mapsto r,\qquad s\mapsto rs
$$
preserves 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 \b[outer automorphism of the nonabelian group $D_8$].