I have a homogeneous Riemannian manifold X with isometry group Iso. Is Iso a maximal group? By maximal group, I mean that there does not exist another group G such that:

1. Iso is a proper subgroup of G,
2. G acts transitively on X by diffeomorphisms, and
3. G has compact stabilizers G_x.

I know that if such a group exists, then there is a G-invariant metric on X. So, in other words, the question is: if I have a Riemannian metric d on X with isometry group Iso which is homogeneous, can there exist another Riemannian metric d' on X with isometry group Iso', such that Iso' contains properly the group Iso?