 Solution

Fix a basepoint $x_0$. A homeomorphism $f:S\to S$ induces an isomorphism from ${\pi }_1(S,x_0)$ to ${\pi }_1(S,f(x_0))$. Choosing a path from $x_0$ to $f(x_0)$ identifies the latter group with $G={\pi }_1(S,x_0)$. Changing the path conjugates the resulting automorphism by an element of $G$, while an isotopy changes nothing in the outer automorphism group. Hence there is a natural homomorphismthe mapping-class action on the outer automorphism group of the fundamental group.