The profinite completion of a group localised at a subgroup

Let $G$ be a group and let $K$ be a commensurated subgroup of $G$. Then there is a totally disconnected, locally compact (t.d.l.c.) group ${\widehat{G}}_K$ that contains the profinite completion of $K$ as an open compact subgroup and also contains $G$ (modulo the finite residual of $K$) as a dense subgroup. Moreover, given an arbitrary group $G$, then every t.d.l.c. group containing an image of $G$ as a dense subgroup can be realised as a quotient of ${\widehat{G}}_K$ for some commensurated subgroup $K$.