On a class of vertex-transitive distance-regular covers of complete graphs, II

Let $\Gamma$ be an abelian antipodal distance-regular graph of diameter 3 with the following property: $(*)$ $\Gamma$ has a transitive group $\overline{G}$ of automorphisms which induces a primitive almost simple permutation group $\overline{G}^{\Sigma}$ on the set ${\Sigma}$ of its antipodal classes. If permutation rank ${\rm rk}(\overline{G}^{\Sigma})$ of $\overline{G}^{\Sigma}$ equals $2$, then $\Gamma$ is arc-transitive; moreover, all such graphs are now known. The purpose of this paper is to describe the graphs $\Gamma$ with the property $(*)$ in the case when ${\rm rk}(\overline{G}^{\Sigma})=3$. According to the classification of primitive almost simple permutation groups of rank $3$ the socle of the group $\overline{G}^{\Sigma}$ under the given condition is either a sporadic simple group, or an alternating group, or a simple group of exceptional Lie type, or a classical simple group. Earlier, we described the graphs $\Gamma$ provided that ${\rm rk}(\overline{G}^{\Sigma})=3$ and the socle of $\overline{G}^{\Sigma}$ is a sporadic simple group. Here we study the cases when $(i)$ the socle of the group $\overline{G}^{\Sigma}$ is an alternating group or $(ii)$ $|{\Sigma}|\le 2500$ and socle of $\overline{G}^{\Sigma}$ is a simple group of exceptional Lie type. We show that the family of non-bipartite graphs $\Gamma$ with the property $(*)$ and $\mathrm{rk}(\overline{G}^{\Sigma})=3$ in the alternating case is finite and limited to a small number of potential examples with $|\Sigma|\in\{10,28,120\}$, each of which is a covering of one of five certain distance-transitive Taylor graphs. For each given group $\overline{G}^{\Sigma}$ of degree $|{\Sigma}|\le 2500$ of exceptional type, we essentially restrict the set of admissible parameters of $\Gamma$.