On intersections of Sylow 2-subgroups in finite groups I

In a finite simple group with simple socle of Lie type over a field of order 3, where Lie rank doesn't exceed 4, we found the lower bound for the number of orbits under the conjugation action of the fixed Sylow 2-subgroup on the set of Sylow 2-subgroups that intersect the fixed on the unit subgroup. Some computer calculations are also used.