Primitive permutation groups with a solvable 2-transitive subconstituent

For a permutation group G acting on a finite set Ω and a point α∈Ω, a suborbit Δ(α) is an orbit of the point stabilizer Gα on Ω. The permutation group induced by Gα on Δ(α) is called a subconstituent of G. Moreover, G is said to be uniprimitive if G is primitive but not 2-transitive. In this paper we investigate uniprimitive permutation groups which have a solvable 2-transitive subconstituent. We determine all such groups G which have a simple socle. The affine case, that is G has an elementary abelian socle, are also discussed and an infinite family of affine primitive groups with non-self-paired 2-transitive subconstituents are presented.