  We investigate a possible connection between the $FSZ$ properties of a group
and its Sylow subgroups. We show that the simple groups $G_2(5)$ and $S_6(5)$,
as well as all sporadic simple groups with order divisible by $5^6$ are not
$FSZ$, and that neither are their Sylow 5-subgroups. The groups $G_2(5)$ and
$HN$ were previously established as non-$FSZ$ by Peter Schauenburg; we present
alternative proofs. All other sporadic simple groups and their Sylow subgroups
are shown to be $FSZ$. We conclude by considering all perfect groups available
through GAP with order at most $10^6$, and show they are non-$FSZ$ if and only
if their Sylow 5-subgroups are non-$FSZ$.
