Working with Dixon and Mortimer's "Permutation Groups", we get explicit descriptions of two-point stabilizers of most of the infinite families of the two-transitive permutation groups. For example, we can readily determine the two-point stabilizers of $AGL(k,n)$, $PGL(k,n)$, $PSU(3,n)$, $Sz(2^{2k+1})$, and $Ree(3^{2k+1})$ on the corresponding affine spaces, projective spaces, isotropic spaces, or Steiner systems.
In the same book, there is a description of the two distinct two-transitive actions of $Sp(2k,2)$, which is well known to be the same as its actions on the sets of cosets of its two maximal orthogonal subgroups. There it is stated that $Sp(2k,2)$ is generated by transvections whenever $k\geq 3$. I tried to work out the form of the matrices in a two-point stabilizer $Sp(2k,2)_{\theta_a,\theta_b}$ through these generators. Even though I determined which transvections would fix these points, my attempt at a characterization failed because products of transvections that do not fix $\theta_a$ and $\theta_b$ may still fix $\theta_a$ and $\theta_b$. I tried to consider when 2 products of transvections would be in the two-point stabilizer, then 3, and so on, but a full characterization remains elusive.
I would greatly appreciate such an explicit description of $Sp(2k,2)_{\theta_a,\theta_b}$, an idea on how to obtain such a description, or a reference from which to work.