Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 156 / 2 / d / Solution

Take the three-punctured sphere $S_{0,3}$. Its full mapping class group is finite, isomorphic to the permutation group $S_3$, whereasand $\mathrm{Out}(F_2)$ is infinite by part c. Hence the finite subgroup $p(\mathrm{Mod}(S_{0,3}))$ has infinite index in $\mathrm{Out}({\pi }_1(S_{0,3}))$.