Inclusions of innately transitive groups into wreath products in product action with applications to $2$-arc-transitive graphs

We study $(G,2)$-arc-transitive graphs for innately transitive permutation groups $G$ such that $G$ can be embedded into a wreath product $\text{sym}\mathrm{\Gamma }\wr \text{sy}\mathrm{\ell }$ acting in product action on ${\mathrm{\Gamma }}^{\mathrm{\ell }}$. We find two such connected graphs: the first is Sylvester’s double six graph with 36 vertices, while the second is a graph with ${120}^2$ vertices whose automorphism group is $\text{aut}\text{sp}44$. We prove that under certain conditions no more such graphs exist.