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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 inthe recent work of Tornier (2018) on local Sylow $p$-subgroups of Burger-Mozesgroups, 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 $(*)_p$. Inthis paper, we extend this result to finite $2$-transitive groups and we give astructural characterisation result for the finite primitive groups that satisfyProperty $(*)_p$ for an allowable prime $p$.

Orbits of Sylow subgroups of finite permutation groups