Finite π-groups with normal injectors

Denote by P the set of all primes and take a nonempty set π ⊆ P. A Fitting class F = (1) is called normal in the class Sπ of all finite soluble π-groups or π-normal, whenever F ⊆ Sπ and for every G ∈ Sπ its F-injectors constitute a normal subgroup of G. We study the properties of π-normal Fitting classes. Using Lockett operators, we prove a criterion for the π-normality of products of Fitting classes. A π-normal Fitting class is normal in the case π = P. The lattice of all solvable normal Fitting classes is a sublattice of the lattice of all solvable Fitting classes; but the question of modularity of the lattice of all solvable Fitting classes is open (see Question 14.47 in [1]). We obtain a positive answer to a similar question in the case of π-normal Fitting classes