Space of $G$-invariant Riemannian metrics contractible?

Question

A well-known result in (psuedo)Riemannian geometry is that the moduli space of (pseudo)Riemannian metrics on a smooth manifold is contractible. In the case when you have a smooth action of a group $G$ on a manifold $M$, is it obvious that the analogous result holds for $G$-invariant Riemannian metrics (say $G$ is a compact Lie group) on $M$? Seems like it should hold... on that note, I suppose a good, additional question would be: are there any any good references on $G$-invariant-type results?