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

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