Intransitive Cartesian decompositions preserved by innately transitive permutation groups

A permutation group is innately transitive if it has a transitive minimal normal subgroup, which is referred to as a plinth. We study the class of finite, innately transitive permutation groups that can be embedded into wreath products in product action. This investigation is carried out by observing that such a wreath product preserves a natural Cartesian decomposition of the underlying set. Previously we classified the possible embeddings in the case where the innately transitive group projects onto a transitive subgroup of the top group. In this article we prove that the transitivity assumption we made in the previous paper was not too restrictive. Indeed, the image of the projection into the top group can only be intransitive when the finite simple group that is involved in the plinth comes from a small list. Even then, the innately transitive group can have at most three orbits on an invariant Cartesian decomposition. A consequence of this result is that if $G$ is an innately transitive subgroup of a wreath product in product action, then the natural projection of $G$ into the top group has at most two orbits.