Generally, I am currently investigating automorphisms of Riemann surfaces and their non-trivial action on the first homology. I wonder whether one can make explicit statements on the relationship between the automorphism group of a surface and the symplectic group, e.g., through the well-known symplectic representation. I recently read about mapping class groups and the natural (surjective) symplectic representation: $\Phi: Mod(S_g)\to Sp(2g, \mathbb{Z})$. In general I am aware that the kernel (the Torelli group) is highly non-trivial, however, I am unsure whether there are cases in which one can make more detailed statements about the relation between an automorphism group of a surface and $Sp(2g, \mathbb{Z})$.
In particular, I wonder if there are any cases in which one can for instance identify 'what part' of the symplectic group we get given the automorphism group of a surface (not necessarily generated by Dehn twists), or even, which elements are needed to generate the full symplectic group. It would for instance be helpful to derive statements like the one exemplified above for hyperbolic surfaces whose automorphism group is some PSL. Is there anything known in this direction, or are there any special cases of surfaces for which deriving statements like the examples above seem doable?
I am aware that in general the action on the homology can be computed numerically but I would like to make analytic statements about potential special cases for which this is possible without having to do numerical computation.