Non-normal orbit quotients that are not normal quotients

Let $G$ be an infinite transitive graph, $\Gamma\le\operatorname{Aut}(G)$ a transitive group and $A\le\Gamma$ a subgroup. Grimmett and Li asked whether $A$ can be non-normal in $\Gamma$ with a transitive orbit quotient $G_A$ that is not isomorphic to $G_N$ for any normal subgroup $N$ of any transitive $\Gamma'\le\operatorname{Aut}(G)$, and whether a free action of $A$ matters. We show that for every $m\ge3$ there is an infinite, connected, locally finite, simple Cayley graph $X$ with a non-normal subgroup $B$ of $\operatorname{Aut}(X)$, acting freely, such that $X_B$ is the complete graph $K_m$, while no normal subgroup of any transitive subgroup of $\operatorname{Aut}(X)$ has $K_m$ as its orbit quotient. The graph is a graphical regular representation of the free product $F*F$, where $F$ is $S_3$ for $m=3$ and the alternating group $A_m$ for $m\ge4$, given by the theorem of Leemann and de la Salle, and $B$ is the preimage of a point stabiliser under the projection to the first factor. The quotient is finite; the question with an infinite quotient is left open.