Distal actions on coset spaces in totally disconnected, locally compact groups

Let $G$ be a totally disconnected, locally compact group and let $H$ be an equicontinuously (for example, compactly) generated group of automorphisms of $G$. We show that every distal action of $H$ on a coset space of $G$ is a SIN action, with the small invariant neighbourhoods arising from open $H$-invariant subgroups. We obtain a number of consequences for the structure of the collection of open subgroups. For example, it follows that for every compactly generated subgroup $K$ of $G$, there is a compactly generated open subgroup $E$ of $G$ such that $K \le E$ and such that every open subgroup of $G$ containing a finite index subgroup of $K$ contains a finite index subgroup of $E$. We also show that for a large class of closed subgroups $L$ of $G$ (including for instance all closed subgroups $L$ such that $L$ is an intersection of subnormal subgroups of open subgroups), every compactly generated open subgroup of $L$ can be realized as $L \cap O$ for an open subgroup of $G$.