On the heritability of the property $D_\pi$ by subgroups

For some set of primes $\pi$, a subgroup $H$ of a finite group $G$ is called a $\pi$-Hall subgroup if all prime divisors of $|H|$ are in $\pi$ and $|G:H|$ has no prime divisors from $\pi$. A group $G$ is said to possess the property $D_\pi$ if it has only one class of conjugate maximal $\pi$-subgroups or, equivalently, the complete analog of Sylow's theorem for Hall $\pi$-subgroups is valid in $G$. We investigate which subgroups of $D_\pi$-groups inherit the property $D_\pi$.