  Let $G$ be a finite group and $\pi$ be a set of primes. We show that if the
number of conjugacy classes of $\pi$-elements in $G$ is larger than $5/8$ times
the $\pi$-part of $|G|$ then $G$ possesses an abelian Hall $\pi$-subgroup which
meets every conjugacy class of $\pi$-elements in $G$. This extends and
generalizes a result of W. H. Gustafson.
