Can you give me an example of a Lie group acting on a compact metric connected space transitively so that it has a closed finite index subgroup which does not act transitively?

I wanted to make this a comment, but it turned out to be too long, so have to submit it as (partial) answer... sorry if I misunderstood the question. Let $G$ be the Lie group. If such an example exists, it would have to be somewhat exotic  by my previous comment, any closed subgroup of finite index has to be a union of connected components of $G$. On the other hand, if a Lie group acts on a differentiable manifold $X$ transitively, then its identity component also acts transitively (see e.g. Onischik and Vinberg, "Lie groups and algebraic groups", Chapter 1 section 3). Hence, to find such an example, one would have to look for $X$ that is not a manifold. On the other hand, if the stabilizer of a point is Lie subgroup, then $X$ would have a unique structure of a manifold with respect to which the action is differentiable. Hence, if you really want such an example, one has to look for some action that "cannot be made smooth" by putting a differentiable manifold structure on $X$, i.e., where the stabilizer of a point is not a submanifold of $G$ (I think traditionally in the "standard" theory of Lie groups such actions are not considered...) 

