2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/71169We study Cartesian decompositions of sets that are acted upon intransitively by innately transitive permutation groups. We prove that such groups have at most three orbits on such a 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 2 orbits.to be submittedGroup Theory05C25, 05C90, 20B05, 20B15, 20B25, 20B35, 20D40text