I have a homogenus Riemannian manifold X with iaometry group Iso. Is Iso a maximal group, I mean that does not exist other group G, such that:
1. Iso is subgroup of G,
2. G act transitivly on X by difeomorphisms,
3. G has compact stabilizers G_x,

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