Intransitive cartesian decompositions preserved by innately transitive permutation groups

A permutation group is innately transitive if it has a transitiveminimal normal subgroup, which is referred to as a plinth. We study the classof finite, innately transitive permutation groups that can be embedded intowreath products in product action. This investigation is carried out by observingthat such a wreath product preserves a natural Cartesian decompositionof the underlying set. Previously we classified the possible embeddings in thecase where the innately transitive group projects onto a transitive subgroupof the top group. In this article we prove that the transitivity assumption wemade in the previous paper was not too restrictive. Indeed, the image of theprojection into the top group can only be intransitive when the finite simplegroup that is involved in the plinth comes from a small list. Even then, theinnately transitive group can have at most three orbits on an invariant Cartesiandecomposition. A consequence of this result is that if G is an innatelytransitive subgroup of a wreath product in product action, then the naturalprojection of G into the top group has at most two orbits.