Let $S$ be an orientable compact surface. A homeomorphism $f: S \to S$ induces an isomorphism $f_{*}: H_1(S) \to H_1(S)$.
How much can we say the converse? Namely, if we are given an element of $\alpha \in$ $\operatorname{Aut}(H_1(S))$, is there a self-homeomorphism $f$ of $S$ (unique up to isotopy or something) such that $f_*=\alpha$?