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

We study (G,2)(G,2)-arc-transitive graphs for innately transitive permutation groups G such that G   can be embedded into a wreath product SymΓwrSℓ acting in product action on ΓℓΓℓ. We find two such connected graphs: the first is Sylvester's double six graph with 36 vertices, while the second is a graph with 1202 vertices whose automorphism group is AutSp(4,4). We prove that under certain conditions no more such graphs exist.