Given a group $G$ and subgroups $U\le V\le G$, there are canonical quotient maps $G/U\to G/V$ that send cosets of $U$ to the cosets of $V$ containing them. Even when $U,V$ aren't normal, this map is a valid morphism in the category of $G$-sets (sets equipped with a $G$-action, the morphisms comprised of bijections that intertwine with the action), so we may form an inverse system of quotients of $G$ using the quotient maps. Since the system contains $G/1$ at the top, the limit is simply $G$, but if we restrict our attention to infinite $G$ and subgroups of finite index, the situation is not so trivial.
Every $G$-set decomposes as a disjoint union of its orbits, orbits are always transitive $G$-sets in their own right, and transitive $G$-sets (leave the empty set out of this) are always isomorphic to quotients of $G$, so it is sensible to ask for the number of orbits of a $G$-set isomorphic to a particular quotient of the group $G$. (This motivates the Burnside ring, I think.) Is there a straightforward way to find this number for the inverse limit of the aforementioned system (or, similarly, of the system formed from quotients by infinite subgroups, which includes finite-index ones but is generally bigger)?
To a first approximation, it might be nice to know when the number is infinite or not, based on whether the $U$ in the quotient $G/U$ is finite or infinite, or the same of its index, or its algebraic relationship to the whole group, etc., and then if this number is ever finite, what exactly it is.
I get the feeling this should be just as simple as the finite case, but nothing really definitive comes to mind, other than perhaps to investigate $\hom(-,G/U)$'s...