How close is $ GL_n (2) $ to being $3$ transitive? The group $ GL_n(2) $ acts transitively on the $ 2^n-1 $ nonzero vectors of $ \mathbb{F}_2^n $. This action is in fact 2-transitive since any pair of distinct nonzero vectors in $ \mathbb{F}_2^n $ is linearly independent and thus for any two pairs there is an element of $ GL_n(2) $ taking one pair to another. How close to being $3$ transitive is this action?
For the case of $ n=3 $ it seems that there are exactly 2 orbits of the action of $ GL_3(2) $ on the $ {7}\choose{3}$$=35 $ triples of distinct nonzero vectors from $ \mathbb{F}_2^3 $. The first orbit is size $28$ and consists of all possible (unordered) bases of $ \mathbb{F}_2^3 $. While the second orbit is size $ 7 $ and consists of all triples whose span has rank $2$ (in other words, triples of the form $ v,w,v+w $).
How does this generalize to $ n > 3 $? Are there always just two orbits for the action on the space of triples, correspond to span having rank 3 versus span with rank 2, or does it get more complicated for large $ n $?
How close is $ GL_n(2) $ to being 3-transitive for large $ n $?
 A: 
Are there always just two orbits for the action on the space of triples, correspond to span having rank 3 versus span with rank 2 ... ?

Yes. Any $3$-subset of $\mathbb{F}_2^n\setminus\{0\}$ is either linearly independent or spans a 2D subspace. (It is impossible to span a 0D subspace because $0$ is omitted, and it is impossible to span a 1D subspace because our scalars are $\mathbb{F}_2$ so every 1D subspace has exactly one nonzero vector.) These are the two orbits.
Given any two linearly independent ordered subsets $\{a,b,c\}$ and $\{a',b',c'\}$, there is a linear transformation defined by extending $a\mapsto a',b\mapsto b',c\mapsto c'$. The stabilizer of (any element of) this orbit is a copy of the affine group $\mathrm{Aff}_{n-3}\mathbb{F}_2$ which may be identified with block upper triangular matrices with lower right block a $3\times 3$ identity matrix. (Take $\{a,b,c\}$ to be $\{e_{n-2},e_{n-1},e_n\}$ for this.)
Meanwhile, any 2D subspace is of the form $\{u,v,u+v\}$. Given another $\{u',v',u'+v'\}$, we know $u,v$ and $u',v'$ are both linearly independent so we can extend $u\mapsto u',v\mapsto v'$ to a linear transformation. The stabilizer of an element is a copy of $\mathrm{Aff}_{n-2}\mathbb{F}_q$ by the same reasoning (take $u=e_{n-1},v=e_n$).
