Radicals in mutually permutable products of finite groups

Let the finite group G=AB be the mutually permutable product of the subgroups A and B and let F be a Fitting class. Then the F-radicals AF and BF of the factors A and B are mutually permutable. Using this, we also prove the inclusion G′∩AFBF⩽GF, which generalizes the fact that A∈F and B∈F implies G′∈F.