I was wondering if anything is known about the possible structure of $\mathrm{Aut}(S)$ for a Riemann surface $S$. More precisely, are there known obstructions for a finite group $G$ to be such an automorphism group?

There are no obstructions.

In fact, every finite group is isomorphic to the full automorphism group $\textrm{Aut}(S)$ of some compact Riemann surface $S$ (of genus at least $2$).

Moreover, $S$ may be chosen so that the quotient Riemann surface $S/\textrm{Aut}(S)$ has any preassigned genus.

For a reference, see Theorem 4 in the paper by L. Greenberg *Maximal Fuchsian groups*, Bull. Amer. Math. Soc. Volume **69**, Number 4 (1963), 569-573.