  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 plinth is simple. Here we extend that classification and identify
several different types of Cartesian decompositions that can be preserved by an
innately transitive group with a non-abelian plinth. These different types of
decompositions lead to different types of embeddings of the acting group into
wreath products in product action. We also obtain a full characterisation of
embeddings of innately transitive groups with diagonal type into such wreath
products.
