Let $G$ be a topological group and let $H$ be a closed subgroup of $G$. Say $H$ is *codistal* in $G$ if the translation action of $G$ on the coset space $G/H$ is distal (meaning that no non-diagonal orbit of $G$ on $G/H \times G/H$ has an accumulation point on the diagonal).

This concept appears (although not with this name) in the paper

H. B. Keynes, 'A study of the proximal relation in coset transformation groups', Trans. Amer. Math. Soc. 128 (1967), 389–402.

Keynes shows that $H$ is codistal in $G$ if and only if $\bigcap_{U \in \mathcal{U}}HUH = H$, where $\mathcal{U}$ is a basis of identity neighbourhoods in $G$. In particular, it is enough to show that the action of $H$ on $G/H$ is distal, and if $K$ is codistal in $H$ and $H$ is codistal in $G$, then $K$ is codistal in $G$.

Beyond that point though I could not find many references. Has this property been studied further under another name? It is a generalization of some other natural situations, for instance any intersection of open subgroups is codistal. I am particularly interested in the following question:

Suppose $G$ is a locally compact group and $H$ is a codistal subgroup of $G$. When does it happen that the action of $H$ on $G/H$ has small invariant neighbourhoods, in other words, $\{HUH/H \mid U \in \mathcal{U}\}$ is a base of neighbourhoods of the trivial coset in $G/H$?

If $G/H$ is compact, it looks like one obtains a small invariant neighbourhoods action by considering the Ellis semigroup of $H$ on $G/H$, so the interesting case is when $G/H$ is not compact. All the examples I know of (at least in totally disconnected, locally compact groups) still satisfy the stronger condition that $H$ has small invariant neighbourhoods on $G/H$, but I suspect this is not true in general.