Finite symmetric graphs with two-arc transitive quotients

This paper forms part of a study of 2-arc transitivity for finite imprimitive symmetric graphs, namely for graphs Γ admitting an automorphism group G that is transitive on ordered pairs of adjacent vertices, and leaves invariant a nontrivial vertex partition B. Such a group G is also transitive on the ordered pairs of adjacent vertices of the quotient graph ΓB corresponding to B. If in addition G is transitive on the 2-arcs of Γ (that is, on vertex triples (α,β,γ) such that α,β and β,γ are adjacent and α≠γ), then G is not in general transitive on the 2-arcs of ΓB, although it does have this property in the special case where B is the orbit set of a vertex-intransitive normal subgroup of G. On the other hand, G is sometimes transitive on the 2-arcs of ΓB even if it is not transitive on the 2-arcs of Γ. We study conditions under which G is transitive on the 2-arcs of ΓB. Our conditions relate to the structure of the bipartite graph induced on B∪C for adjacent blocks B,C of B, and a graph structure induced on B. We obtain necessary and sufficient conditions for G to be transitive on the 2-arcs of one or both of Γ,ΓB for certain families of imprimitive symmetric graphs.