On primitive overgroups of quasiprimitive permutation groups

A permutation group is said to be quasiprimitive if all its non-trivial normal subgroups are transitive. We investigate pairs (G,H) of permutation groups of degree n such that G H Sn with G quasiprimitive and H primitive. An explicit classification of such pairs is obtained except in the cases where the primitive group H is either almost simple or the blow-up of an almost simple group. The theory in these remaining cases is investigated in separate papers. The results depend on the finite simple group classification.