  In the vein of Bonfert-Taylor, Bridgeman, Canary, and Taylor we introduce the
notion of quasiconformal homogeneity for closed oriented hyperbolic surfaces
restricted to subgroups of the mapping class group. We find uniform lower
bounds for the associated quasiconformal homogeneity constants across all
closed hyperbolic surfaces in several cases, including the Torelli group,
congruence subgroups, and pure cyclic subgroups. Further, we introduce a
counting argument providing a possible path to exploring a uniform lower bound
for the nonrestricted quasiconformal homogeneity constant across all closed
hyperbolic surfaces.
