Let $R$ be a compact Riemann surface of genus $g$ and let $S \subset R$ be a Riemann subsurface. Theorem B in Maskit's paper says that we can embed $S$ into a compact Riemann surface $P$ of genus $g$ in such a way that conformal automorphisms of the image extend to be conformal automorphisms of $P$.
Theorem A in the same paper is the analogous result for planar domains. I would like to know if Maskit's result is easier to prove if we assume that the Riemann surface $S$ is finite, i.e., if the fundamental group of $S$ is finitely generated. Note that for planar domains, Maskit's Theorem A for domains with finitely generated fundamental group follows from Keobe's generalisation of the Riemann mapping theorem to finitely-connected domains.
