On one generalization of $\sigma$-local and Baer-local formations

Throughout this paper, all groups are finite and $G$ is a group. Let $\sigma=\{\sigma_i\mid i\in I\}$ be some partition of the set of all primes $\mathbb{P}$. Then $\sigma(G)=\{\sigma_i\mid \sigma_i\cap\pi(G)\ne\varnothing\}$; $\sigma^+(G)=\{\sigma_i\mid G \text{ has a chief factor } H/K, \text{ such that } \sigma(H/K)=\{\sigma_i\}\}$. The group $G$ is said to be: $\sigma$-primary if $G$ is $\sigma_i$-group for some $i$; $\sigma$-soluble if every chief factor of $G$ is $\sigma$-primary. The symbol $R_\sigma(G)$ denotes the product of all normal $\sigma$-soluble subgroups of $G$. The chief factor $H/K$ of $G$ is said to be: $\sigma$-central (in $G$) if $(H/K)\rtimes(G/C_G(H/K))$ is $\sigma$-primary; a $\sigma_i$-factor if $H/K$ is a $\sigma_i$-group. We say that $G$ is: $\sigma$-nilpotent if every chief factor of $G$ is $\sigma$-central; generalized $\{\sigma_i\}$-nilpotent if every chief $\sigma_i$-factor of $G$ is $\sigma$-central. The symbol $F_{\{g\sigma_i\}}(G)$ denotes the product of all normal generalized $\{\sigma_i\}$-nilpotent subgroups of $G$. We call any function $f$ of the form $f:\sigma\cup\{\varnothing\}\to\{\text{formations of groups}\}$, where $f(\varnothing)\ne\varnothing$, a generalized formation $\sigma$-function and we put
$$BLF_\sigma(f)=(G\mid G/R_\sigma(G)\in f(\varnothing) \text{ and } G/F_{\{g\sigma_i\}}(G)\in f(\sigma_i) \text{ for all }\sigma_i\in\sigma^+(G)).$$
If for some generalized formation $\sigma$-function $f$ we have $\mathfrak{F}=BLF_\sigma(f)$, then we say that the class $\mathfrak{F}$ is Baer-$\sigma$-local and $f$ is a generalized $\sigma$-local definition of $\mathfrak{F}$. In this paper, we describe basic properties, examples, and some applications of Baer-$\sigma$-local formations.