Let $G=\langle g_1,g_2\rangle$ denote the free group of rank 2. For a subgroup $H$ of $G$ consider the quotient graph with vertex set $H\setminus G$ of right cosets, where $Hg$ and $Hg'$ are connected by an edge if $Hg g_i^{\pm 1}=Hg'$ for $i=1$ or $i=2$.

Does there exist a transitive action on $H\setminus G$, which commutes with each of the mappings given by right-multiplication with $g_1^{\pm 1 }$ or $g_2^{\pm 1}$ on $H\setminus G$?

Clearly, the answer is yes if $H\setminus G$ is a group (and the action is given by left-multiplication). Can you think of other examples or related concepts?

EDIT: Unfortunately, I don't understand R W's answer. It seems that he says that - in general - the answer to my question is "no". (Please let me know if I'm wrong.)

On the other hand, the answer is "yes" for all normal subgroups (left and right multiplication commute). So, for which subgroups is the answer "yes"?