I was doing something which needs to know sizes of all orbits of the stabilizer of two points in a 2-transitive permutation group. Since all 2-transitive permutation groups are known ans so are their stabilizer of two points, it is not to hard to find all the sizes of all orbits of these stabilizers. However, it seems to me that this case by case discuss will not be a short argument, so I would like to ask: are there any known results (for some or all infinite families of 2-transitive permutation groups)?

I was doing something which needs to know sizes of all orbits of the stabilizer of two points in a 2-transitive permutation group. Since all 2-transitive permutation groups are known ans so are their stabilizer of two points, it is not to hard to find all the sizes of all orbits of these stabilizers. However, it seems to me that this case by case discuss will not be a short argument, so I would like to ask: are there any known results? In particular, are there formulas for $PSU(3,q).[de]$, $Sz(q).e$ and $Ree(q).e$, where $q=p^f$, $d\mid gcd(3,q+1)$ and $e\mid f$?