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: \(G=A\rtimes B\) $G=A⋊B$ , 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.