  We say that a finite group $G$ acting on a set $\Omega$ has Property $(*)_p$
for a prime $p$ if $P_\omega$ is a Sylow $p$-subgroup of $G_\omega$ for all
$\omega\in\Omega$ and Sylow $p$-subgroups $P$ of $G$. Property $(*)_p$ arose in
the recent work of Tornier (2018) on local Sylow $p$-subgroups of Burger-Mozes
groups, and he determined the values of $p$ for which the alternating group
$A_n$ and symmetric group $S_n$ acting on $n$ points has Property $(*)_p$. In
this paper, we extend this result to finite $2$-transitive groups and we give a
structural characterisation result for the finite primitive groups that satisfy
Property $(*)_p$ for an allowable prime $p$.
