Pronormality of Hall subgroups in their normal closure

It is known that for any set $\pi$ of prime numbers, the following assertions are equivalent:
(1) in any finite group, $\pi$-Hall subgroups are conjugate;
(2) in any finite group, $\pi$-Hall subgroups are pronormal.
It is proved that (1) and (2) are equivalent also to the following:
(3) in any finite group, $\pi$-Hall subgroups are pronormal in their normal closure.
Previously [Unsolved Problems in Group Theory, The Kourovka Notebook, 18th edn., Institute of Mathematics SO RAN, Novosibirsk (2014), Quest. 18.32], the question was posed whether it is true that in a finite group, $\pi$-Hall subgroups are always pronormal in their normal closure. Recently, M. N. Nesterov [Sib. El. Mat. Izv., 12 (2015), 1032–1038] proved that assertion (3) and assertions (1) and (2) are equivalent for any finite set $\pi$. The fact that there exist examples of finite sets $\pi$ and finite groups $G$ such that $G$ contains more than one conjugacy class of $\pi$-Hall subgroups gives a negative answer to the question mentioned. Our main result shows that the requirement of finiteness for $\pi$ is unessential for (1), (2), and (3) to be equivalent.