Alice Devillers, Cai Heng Li, Geoff Pearce, Cheryl Praeger and I have just uploaded to the arxiv a preprint of our recently submitted paper `On imprimitive rank 3 permutation groups‘.
A permutation group on a set also acts on the set via . If is transitive on then we cannot expect it to be transitive on as the subset is an orbit. The best that we can hope for is that has two orbits on and this is equivalent to acting 2-transitively on , that is, acts transitively on the set of ordered pairs of distinct elements of . The rank of a permutation group is the number of orbits of on . Thus 2-transitive groups have rank 2.
The number of orbits of on is equal to the number of orbits of the point stabiliser on . Indeed, given an orbit of on the set is an orbit of on . Conversely, given an orbit of on , the set is an orbit of on . In this set up, corresponds to .