Finite 2-arc-transitive strongly regular graphs and 3-geodesic-transitive graphs

We classify all the $2$-arc-transitive strongly regular graphs, and use this classification to study the family of finite $(G,3)$-geodesic-transitive graphs of girth $4$ or $5$ for some group $G$ of automorphisms. For this application we first give a reduction result on the latter family of graphs: let $N$ be a normal subgroup of $G$ which has at least $3$ orbits on vertices. We show that $\Gamma$ is a cover of its quotient $\Gamma_N$ modulo the $N$-orbits, and that either $\Gamma_N$ is $(G/N,3)$-geodesic-transitive of the same girth as $\Gamma$, or $\Gamma_N$ is a $(G/N,2)$-arc-transitive strongly regular graph, or $\Gamma_N$ is a complete graph with $G/N$ acting 3-transitively on vertices. The classification of $2$-arc-transitive strongly regular graphs allows us to characterise the $(G,3)$-geodesic-transitive covers $\Gamma$ when $\Gamma_N$ is complete or strongly regular.