The classification of doubly transitive groups with simple socle is known. A good account of such classification can be found for example in this paper:

Cameron, Peter J. Finite permutation groups and finite simple groups. Bull. London Math. Soc. 13 (1981), no. 1, 1--22. MR0599634 (83m:20008)

Another reference is Chapter 7 (Section 7.7) of

Dixon, John D.; Mortimer, Brian. Permutation groups. Graduate Texts in Mathematics, 163. Springer-Verlag, New York, 1996. xii+346 pp. ISBN: 0-387-94599-7 MR1409812 (98m:20003)

I am particularly interested in the following permutation groups of the classification: $PSL(d,q)$ with $d>2$, $Sp(2d,2)$ with $d>2$, $PSU(3,q)$ with $q>2$, $Sz(q)$ (Suzuki groups) with $q^{2n+1}>2$ and $Ree(q)$ (Ree groups) with $q^{3n+1}>3$.

Let $G$ be any of these groups and suppose that $G$ acts $2$-transitively on a set $\Omega$. My question is the following:

Which of these permutation groups are doubly primitive on $\Omega$? (Doubly primitive means that the one-point stabilizer $G_\alpha$ is primitive on $\Omega\setminus\{\alpha\}$.)

**Motivation**: I need to prove that the centers of the one-point stabilizers of the groups I mentioned are trivial. This can be done case-by-case. (Unfortunately, I cannot find a general argument.) I realized that the calculations of the centers could be avoided in the cases where $G$ is doubly primitive on $\Omega$. (This happens for example when $G=Sp(2d,2)$, see this MO Question.)