On Lockett pairs and Lockett conjecture for $\sigma$-local Fitting classes

For each nonempty Fitting class $\mathfrak{F}$, Lockett defined the smallest Fitting class $\mathfrak{F}^*$ containing $\mathfrak{F}$ such that $(G\times H)_{\mathfrak{F}^*}=G_{\mathfrak{F}^*}\times H_{\mathfrak{F}^*}$ for all groups $G$ and $H$ and the Fitting class $\mathfrak{F}_*$ as the intersection of all nonempty Fitting classes $\mathfrak{X}$ for which $\mathfrak{X}^*=\mathfrak{F}^*$. Lockett pair of nonempty Fitting classes $\mathfrak{F}$ and $\mathfrak{H}$ is an ordered pair $(\mathfrak{F},\mathfrak{H})$ such that $\mathfrak{F}\cap\mathfrak{H}_*=(\mathfrak{F}\cap\mathfrak{H})_*$. If $\mathfrak{F}\subseteq\mathfrak{H}$ and $\mathfrak{F}$ is a Lockett class, then $\mathfrak{F}$ is said to satisfy Lockett conjecture in $\mathfrak{H}$. In the present paper, in the universe $\mathfrak{S}$ of all finite soluble groups, the methods for constructing Lockett pairs are described for the case when $\mathfrak{F}$ is a generalized local Fitting class, and, in particular, for $\mathfrak{F}$ confirmed Lockett conjecture.