Quotient-complete arc-transitive latin square graphs from groups

We consider latin square graphs $\mathrm{\Gamma }=\mathrm{L}\mathrm{S}\mathrm{G}(H)$ of the Cayley table of a given finite group $H$. We characterize all pairs $(\mathrm{\Gamma },G)$, where $G$ is a subgroup of autoparatopisms of the Cayley table of $H$ such that $G$ acts arc-transitively on $\mathrm{\Gamma }$ and all nontrivial $G$-normal quotient graphs of $\mathrm{\Gamma }$ are complete. We show that $H$ must be elementary abelian and determine the number $k$ of complete normal quotients. This yields new infinite families of diameter two arc-transitive graphs with $k=1$ or $k=2$.