The central problem asks whether finite tournaments, which are complete directed graphs where every pair of vertices has exactly one directed edge between them, have a strong extension property with respect to symmetric groups. Specifically, one wants to know if every finite tournament can be represented inside a finite symmetric group such that any partial permutation consistent with the tournament's structure can always be extended to a full permutation of the group. This is closely tied to the long-standing open question of whether the class of finite tournaments has the extension property for partial automorphisms, meaning whether partial automorphisms of any finite tournament always extend to automorphisms of some larger finite tournament. The paper on extension properties for partial permutations circles around precisely this kind of question without fully resolving it in the tournament setting.