Orbits of Sylow subgroups of finite permutation groups

We say that a finite group G acting on a set Ω has Property (⁎)_p for a prime p if P_ω is a Sylow p-subgroup of G_ω for all ω∈Ω 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.