Finite groups with given systems of generalised $\sigma$-permutable subgroups

Let $\sigma = {\sigma_i | i \in I}$ – be a partition of the set of all primes $\mathbb{P}$ and $G$ – be a finite group. A set $\mathbb{P}$ of subgroups of $G$ is said to be a complete Hall $\sigma$-set of $G$ if every member $\ne 1$ of $\mathcal{H}$ is a Hall $\sigma_i$-subgroup of $G$ for some $i \in I$ and $\mathcal{H}$ contains exactly one Hall $\sigma_i$-subgroup of $G$ for every $i$ such that $\sigma_i\cap\pi(G)\ne\oslash$. A group is said to be $\sigma$-primary if it is a finite $\sigma_i$-group for some $i$. A subgroup $A$ of $G$ is said to be: $\sigma$-permutable in $G$, if $G$ possesses a complete Hall $\sigma$-set $\mathcal{H}$ such that $AH^x = H^xA$ for all $H \in \mathcal{H}$ and all $x \in G$; $\sigma$-subnormal in $G$, if there is a subgroup chain $A = A_0 \leq A_1\leq\ldots\leq A_t = G$ such that either $A_{i-1} \trianglelefteq A_i$, or $A_i/( A_i - 1)_{A_i}$ is $\sigma$-primary for all $i = 1,\ldots, t$; $\mathfrak{U}$-normal in $G$ if every chief factor of $G$ between $A_G$ and $A^G$ is cyclic. We say that a subgroup $H$ of $G$ is: (i) partially $\sigma$-permutable in $G$ if there are $\mathfrak{U}$-normal subgroup $A$ and a $\sigma$-permutable subgroup $B$ of $G$ such that $H = < A, B >$; (ii) $(\mathfrak{U}, \sigma)$-embedded in Gif there are a partially $\sigma$-permutable subgroup $S$ and a $\sigma$-subnormal subgroup $T$ of $G$ such that $G = HT$ and $H \cap T \leq S \leq H$. We study $G$ assuming that some subgroups of $G$ are partially $\sigma$-permutable or $(\mathfrak{U}, \sigma)$-embedded in $G$. Some known results are generalised.