Sylow properties of finite groups

Let $\mathfrak{F}$ be a non-empty class of finite groups, and $\pi$ be some set of prime numbers. An $S_\pi$-subgroup of group $G$ that belongs to the class $\mathfrak{F}$ is called an $S_\pi(\mathfrak{F})$-subgroup of $G.$ $C_\pi(\mathfrak{F})$ is the class of all groups $G$ that have $S_\pi(\mathfrak{F})$-subgroups, and any two $S_\pi(\mathfrak{F})$-subgroups of $G$ are conjugate in $G;$ $D_\pi(\mathfrak{F})$ is the class of all $C_\pi(\mathfrak{F})$-groups $G$ in which every $\mathfrak{F}_\pi$-subgroup is contained in some $S_\pi(\mathfrak{F})$-subgroup of $G.$ In this paper the new D-theorems are obtained, a number of properties of $D_\pi(\mathfrak{F})$-groups, and $C_\pi(\mathfrak{F})$-groups are established.