On the pronormality of Hall subgroups

Fix a set of primes $\pi$. A finite group is said to satisfy $C_\pi$ or, in other words, to be a $C_\pi$-group, if it possesses exactly one class of conjugate $\pi$-Hall subgroups. The pronormality of $\pi$-Hall subgroups in $C_\pi$-groups is proven, or, equivalently, we show that $C_\pi$ is inherited by overgroups of $\pi$-Hall subgroups. Thus an affirmative solution is obtained to Problem 17.44(a) from The Kourovka Notebook. We also provide some example demonstrating that Hall subgroups in finite groups are not pronormal in general.