Mg.metric geometry – evenly uniform homogeneous spaces

Let's say we have a homogeneous space $ H setminus G $.

Is it possible to say whether this homogeneous space admits a metric conforming to the structure only from its group structure?

I am particularly interested in a situation where $ H $ is a noncompact subgroup at most $ G $.

I hope my question does not seem too broad. Maybe this question has a trivial answer, but it's not obvious to a theoretical physicist.