Two-geodesic transitive graphs of prime power order

In a non-complete graph Γ, a vertex triple (u,v,w) with v adjacent to both u and w is called a 2-geodesic if uneqw and u,w are not adjacent. The graph Γ is said to be 2-geodesic transitive if its automorphism group is transitive on arcs, and also on 2-geodesics. We first produce a reduction theorem for the family of 2-geodesic transitive graphs of prime power order. Next, we classify such graphs which are also vertex quasiprimitive.