The study of factorized groups has played an important role in the theory of groups. We can consider so relevant results as the Ito's Theorem about products of abelian groups or the celebrated Theorem of Kegel-Wielandt about the solubility of a product of two nilpotent groups. In the very much special case when the factors are normal and nilpotent, a well-known result due to Fitting shows that the product is nilpotent. Nevertheless it is not true in general that the product of two normal supersoluble subgroups of a group is a supersoluble group. To create intermediate situations it is usual to consider products of groups whose factors satisfy certain relations of permutability. Following Carocca [12] we say that a group G = AB is the mutually permutable product of A and B if A permutes with every subgroup of B and vice versa. If, in addition, every subgroup of A permutes with every subgroup of B, we say that the group G is a totally permutable product of A and B.

In this context, we can consider as seminal the following results of Asaad and Shaalan.

1. Theorem A (Asaad and Shaalan [2]) (i) Assume that a group G = AB is the mutually permutable product of A and B. Suppose that A and B are supersoluble and that either A, B or G', the derived subgroup of G, is nilpotent. Then G is supersoluble.

2. […]