On the heritability of the Sylow $\pi$-theorem by subgroups

Let $\pi$ be a set of primes. We say that the Sylow $\pi$-theorem holds for a finite group $G$, or $G$ is a $\mathscr D_\pi$-group, if the maximal $\pi$-subgroups of $G$ are conjugate. Obviously, the Sylow $\pi$-theorem implies the existence of $\pi$-Hall subgroups. In this paper, we give an affirmative answer to Problem 17.44, (b), in the Kourovka notebook: namely, we prove that in a $\mathscr D_\pi$-group an overgroup of a $\pi$-Hall subgroup is always a $\mathscr D_\pi$-group.
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