Suppose that $G$ is a profinite group with the property that every open compact subgroup is topologically finitely generated and just infinite. Suppose that $H$ is a commensurated subgroup of $G$ with countably infinite index. I am interested in whether such subgroups can exist and what is known about them, in particular whether there are any sufficient conditions on $G$ to rule out their existence. Any useful reading material would be much appreciated.
Let me make two comment: If $G$ is finitely generated, than every open subgroup is finitely generated (Edit: I removed the reference). If every open subgroup is just infinite, then the group is called hereditarily just infinite.
A subgroup of countable index cannot be closed in a compact group, e.g. profinite group, because it cannot be measurable.