Let $\mathfrak{F}$ be a class of finite groups and $G$ a finite group. Let ${\mathcal{L}}_{\mathfrak{F}}(G)$ be the set of all subgroups $A$ of $G$ with $A^{G}/A_{G}\in \mathfrak{F}$. A chief factor $H/K$ of $G$ is $\mathfrak{F}$-central in $G$ if $(H/K)\rtimes (G/C_{G}(H/K))\in \mathfrak{F}$. We study the structure of $G$ under the hypothesis that every chief factor of $G$ between $A_{G}$ and $A^{G}$ is $\mathfrak{F}$-central in $G$ for every subgroup $A\in {\mathcal{L}}_{\mathfrak{F}}(G)$. As an application, we prove that a finite soluble group $G$ is a PST-group if and only if $A^{G}/A_{G}\leq Z_{\infty }(G/A_{G})$ for every subgroup $A\in {\mathcal{L}}_{\mathfrak{N}}(G)$, where $\mathfrak{N}$ is the class of all nilpotent groups.

Suppose that $G$ is a finite group and $H$ is a subgroup of $G$. $H$ is said to be $s$-semipermutable in $G$ if $H{{G}_{p}}\,=\,{{G}_{p}}H$ for any Sylow $p$-subgroup ${{G}_{p}}$ of $G$ with $\left( p,\,\left| H \right| \right)\,=\,1$; $H$ is said to be $s$-quasinormally embedded in $G$ if for each prime $p$ dividing the order of $H$, a Sylow $p$-subgroup of $H$ is also a Sylow $p$-subgroup of some $s$-quasinormal subgroup of $G$. In every non-cyclic Sylow subgroup $P$ of $G$ we fix some subgroup $D$ satisfying $1\,<\,\left| D \right|\,<\,\left| P \right|$ and study the structure of $G$ under the assumption that every subgroup $H$ of $P$ with $\left| H \right|\,=\,\left| D \right|$ is either $s$-semipermutable or $s$-quasinormally embedded in $G$. Some recent results are generalized and unified.

A subgroup H is called weakly s-permutably embedded in G if there are a subnormal subgroup T of G and an s-permutably embedded subgroup Hse of G contained in H such that G=HT and H∩T≤Hse. In this note, we study the influence of the weakly s-permutably embedded property of subgroups on the structure of G, and obtain the following theorem. Let ℱ be a saturated formation containing 𝒰, the class of all supersolvable groups, and G a group with E as a normal subgroup of G such that G/E∈ℱ. Suppose that P has a subgroup D such that 1<∣D∣<∣P∣ and all subgroups H of P with order ∣H∣=∣D∣ are s-permutably embedded in G. Also, when p=2 and ∣D∣=2 , we suppose that each cyclic subgroup of P of order four is weakly s-permutably embedded in G. Then G∈ℱ.

A subgroup A of a group G has the strong cover-avoidance property in G, or A is a strong CAP-subgroup of G, if A either covers or avoids every chief factor of every subgroup of G containing A. The main aim of the present paper is to analyse the impact of the strong cover and avoidance property of the members of some relevant families of subgroups on the structure of a group.