Finite $\pi$-groups with normal injectors

Denote by $\mathbb P$ the set of all primes and take a nonempty set $\varnothing\ne\pi\subseteq\mathbb P$. A Fitting class $\mathfrak F\ne(1)$ is called normal in the class $\mathfrak S_\pi$ of all finite soluble $\pi$-groups or $\pi$-normal, whenever $\mathfrak{F\subseteq S}_\pi$ and for every $G\in\mathfrak S_\pi$ its $\mathfrak F$-injectors constitute a normal subgroup of $G$.
We study the properties of $\pi$-normal Fitting classes. Using Lockett operators, we prove a criterion for the $\pi$-normality of products of Fitting classes. A $\pi$-normal Fitting class is normal in the case $\pi=\mathbb P$. 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 $\pi$-normal Fitting classes.