On $n$-multiply $\sigma$-locality of a non-empty $\tau$-cloused formation of finite groups

All groups under consideration are finite. Let $\sigma =\{\sigma_{i} \mid i\in I \}$ be some partition of the set of all primes, $G$ be a group, $\sigma (G)=\{\sigma_i\mid \sigma_i\cap \pi (G)\ne \varnothing\}$, $\mathfrak F$ be a class of groups, and $\sigma (\mathfrak{F})=\bigcup\limits_{G\in \mathfrak{F}}\sigma (G).$ A function $f$ of the form $f:\sigma \to\{\text{formations of groups}\}$ is called a formation $\sigma$-function. For any formation $\sigma$-function $f$ the class $LF_{\sigma}(f)$ is defined as follows: $LF_{\sigma}(f)=(G \mid G=1 \ \text{or }\ G\ne 1\ \text{and }G/O_{\sigma_i', \sigma_i}(G) \in f(\sigma_{i}) \ \text{ for all } \sigma_i \in \sigma(G)).$ If for some formation $\sigma$-function $f$ we have $\mathfrak{F}=LF_{\sigma}(f),$ then the class $\mathfrak{F}$ is called $\sigma$-local and $f$ is called a $\sigma$-local definition of $\mathfrak{F}.$ Every formation is called $0$-multiply $\sigma$-local. For $n \geqslant 1,$ a formation $\mathfrak{F}$ is called $n$-multiply $\sigma$-local provided either $\mathfrak{F}=(1)$ is the class of all identity groups or $\mathfrak{F}=LF_{\sigma}(f),$ where $f(\sigma_i)$ is $(n-1)$-multiply $\sigma$-local for all $\sigma_i\in \sigma (\mathfrak{F}).$ Let $\tau(G)$ be a set of subgroups of $G$ such that $G\in \tau(G).$ Then $\tau$ is called a {subgroup functor} if for every epimorphism $\varphi$ : $A \to~B$ and any groups $H \in \tau (A)$ and $T\in \tau (B)$ we have $H^{\varphi}\in\tau(B)$ and $T^{{\varphi}^{-1}}\in\tau(A).$ A class of groups $\mathfrak{F}$ is called {$\tau$-closed} if $\tau(G)\subseteq\mathfrak{F}$ for all $G\in\mathfrak F.$ In this paper, necessary and sufficient conditions for $n$-multiply $\sigma$-locality $(n\geqslant 1)$ of a non-empty $\tau$-closed formation are obtained.