  We obtain a compact Sobolev embedding for $H$-invariant functions in compact
metric-measure spaces, where $H$ is a subgroup of the measure preserving
bijections. In Riemannian manifolds, $H$ is a subgroup of the volume preserving
diffeomorphisms: a compact embedding for the critical exponents follows. The
results can be viewed as an extension of Sobolev embeddings of functions
invariant under isometries in compact manifolds.
