Let $G \subset \rm{PSL}_{2}\mathbb{C}$ be a subgroup of the Mobius group of the 2-sphere $S^2$, and suppose that $G$ also acts on a second surface $M^2$ by automorphisms.
Does there exist a meromorphic function $f : M^2 \to S^2$ such that $f \circ g = g \circ f$ for all $g \in G$?
When $G$ is finite and $M^2 = S^2$, Doyle and McMullen described the space of such functions in https://math.dartmouth.edu/~doyle/docs/icos/icos/icos.html. Brady did the same much earlier for $G = \rm{PSL}_{2}\mathbb{Z}$ and $M^2 = H^2$ the upper half-plane in http://www.ams.org/journals/proc/1971-030-02/S0002-9939-1971-0280712-5/ (and his method generalizes to other congruence subgroups). I would like to know if anything similar is known in the more general setting. I am particularly interested in the case when $M^2$ is compact of genus greater than one (in which case $G$ is again finite), since it is not even clear (to me) whether such a function exists.
