It is well known that Sobolev embeddings can be refined in the presence of symmetries. Hebey and Vaugon (1997) studied this phenomena in the context of an arbitrary Riemannian manifold M and a compact group of isometries G. They showed that the limit Sobolev exponent increases if there are no points in M with discrete orbits under the action of G. In the paper, the situation where M contains points with discrete orbits is considered. It is shown that the limit Sobolev exponent for (Formula Presented) increases in the case of embeddings into weighted spaces L_q(M, w) instead of the usual L_q spaces, where the weight function w(x) is a positive power of the distance from x to the set of points with discrete orbits. Also, embeddings of (Formula Presented) into weighted Hölder and Orlicz spaces are treated.