Fitting Classes with Given Properties of Hall Subgroups

In the theory of formations of finite solvable groups, there is a well-known result due to Blessenohl claiming that, for any local formation $\mathfrak F$, the class of groups for which every Hall $\pi$-subgroup belongs to $\mathfrak F$ also is a local formation. In the present paper, we obtain a result exactly dual to that indicated in the theory of Fitting classes. We prove that if a Fitting class $\mathfrak F$ is local, then the class of all groups all of whose Hall $\pi$-subgroups belong to $\mathfrak F$ is also local.