Determining hypercentral Hall subgroups in finite groups

Let $G$ be a finite group, and let $\pi$ be a set of primes. The aim of this paper is to obtain some results concerning how much information about the $\pi$-structure of $G$ can be gathered from the knowledge of the lengths of conjugacy classes of its $\pi$-elements and of their multiplicities. Among other results, we prove that this multiset of class lengths determine whether $G$ has a hypercentral Hall $\pi$-subgroup.