On the almost simple automorphism groups of rank 3 graphs

A permutation group $G$ of a finite set $\Omega$ acts componentwisely on the Cartesian square $\Omega^2$. The largest subgroup of $\operatorname{Sym}(\Omega)$ having the same orbits on $\Omega^2$ as $G$ is called the $2$-closure of $G$. The rank of $G$ is the number of its orbits on $\Omega^2$. If the rank of $G$ is $3$ and the order is even, then an undirected graph with vertex set $\Omega$ is defined up to taking complement, for which one of the two off-diagonal orbits of $G$ on $\Omega^2$ is taken as the edge set. Such a graph is called a graph of rank $3$. The full automorphism group of this graph coincides with the $2$-closure of $G$ and contains $G$ as a subgroup. At present, except for the case when $G$ is an almost simple group, there is an explicit description of the $2$-closures of groups $G$ of rank $3$. In this paper, we fill the existing gap, thereby completing the description of the complete automorphism groups of graphs of rank $3$.