I am looking for information about the symplectic groups $Sp_{2m}(2)$ as permutation group acting on quadratic forms.

Consider the block matrices

$$e=\begin{pmatrix}0&1\\0&0\end{pmatrix}, \qquad f=\begin{pmatrix}0&1\\-1&0\end{pmatrix}=e-e^T$$

on the vector space $(\mathbb{F}_2)^{2d}$ equipped with the standard basis. Consider the symmetric bilinear form $\phi(u,v)=ufv^T$, and let $\Omega$ be the set of all quadratic forms $\theta(u)$ such that

$$\phi(u,v)=\theta(u+v)-\theta(u)-\theta(v).$$

In particular the quadratic form $\theta_0(u)=ueu^T$ is $\in\Omega$, and any other element of $\Omega$ can be shown to be of the form

$$\theta_a = \theta_0(u)+\phi(u,a).$$

Now $Sp_{2m}(2)$ acts on $\Omega$, and it turns out that the action splits in two distinct orbits

$$\Omega^+=\{\theta_a|\theta_0(a)=0\},\qquad \Omega^-=\{\theta_a|\theta_0(a)=1\},$$

of size respectively $2^{m-1}(2^m+1)$ and $2^{m-1}(2^m-1)$. The group $Sp_{2m}(2)$ acts $2$-transitively on each of these orbits, see Chap. 7 of *Permutation Groups (Dixon, Mortimer)* for more details.

**Question**: what can be said of the action of these two sets? Here are two more specific questions: is the stabilizer of one point acting imprimitevely, for some block structure? What are the orbits of a $2$-point stabilizer?

**Motivation**: I am studing the Galois groups of polynomials (trinomials) over function fields in characteristic $p$, which can be proven to be $2$-transitive. This was done by Abhyankar, *Galois theory on the line in nonzero characteristic* (1996), which computed the Galois group of many trinomials, and I think that his results can be extended to cover more cases. If I'm wrong I will happen to have learned something about $2$-transitive groups.

The $2$-transitive permutations groups are classified (affine groups, alternating/symmetric, projective, symplectic $Sp_{2m}(2)$, unitary $PGU_3(q^3)$, Suzuki $Sz(q)$ and Ree $R(q)$, plus a few sporadic groups). Computing the local Galois group at a ramified place it is possible to describe the action of a subgroup, the inertia subgroup, as a permutation group on the roots. This allows to rule out certain familes of $2$-transitive groups, and sometimes it is possible to determine completely the Galois group. And the symplectic group at the moment is the family that I find more difficult to understand.