An infinite family of biquasiprimitive 2-arc transitive cubic graphs
  A new infinite family of bipartite cubic 3-arc transitive graphs is constructed and studied. They provide the first known examples admitting a 2-arc transitive vertex-biquasiprimitive group of automorphisms for which the index two subgroup fixing each half of the bipartition is not quasiprimitive on either bipartite half. 
Alice Devillers, Michael Giudici, Cai Heng Li and Cheryl E. Praeger
math.GR