The formation residual of factorized finite groups

Let G be a finite group and G be a split extension of A by B , that is, G is a semidirect product: G = A ⋊ B , where A and B are subgroups of G . Under the condition that B permutes with every maximal subgroup of Sylow subgroups of A , every maximal subgroup of A or every nontrivial normal subgroup of A , we prove that the supersolvable residual of G is the product of the supersolvable residuals of A and B .