Action of a Lie group on a coset of its subgroup

Question

I am a physicist, so sorry for the lack of rigor. It is well known that a (say compact) Lie group $G$ acts naturally by left multiplication on the coset space $G/H$ where $H\subset G$ is its (Lie) subgroup. Suppose now that $\tilde G$ is a subgroup of $G$ and define $\tilde H=H\cap\tilde G$. We can assume (this is the case that I am actually interested in) that $\tilde G$ is a centralizer of a given one-parameter subgroup of $G$. How, and under what conditions, can one define a (transitive) action of $G$ on the coset space $\tilde G/\tilde H$?