I'm attempting exercise 2.8.14 of Dixon & Mortimer. It asks you to show that for $d\geq 1$, $AGL_d(F)$ contains a sharply 2-transitive subgroup $H$. For $d=1$ this is easy, since $AGL_1(F)$ itself is sharply 2-transitive. I wanted to first look at the order in the finite field case to get some ideas. Since $H$ is sharply 2-transitive, its order is necessarily $q^d(q^d-1)$, therefore its index in $AGL_d(F)$ is $|AGL_d(F):H|=q\cdot|AGL_{d-1}|$. I thought that $H$ could be a normal subgroup and I am trying to find an epimorphism from $AGL_d(F)$ into its subgroup $F\times AGL_{d-1}(F)$ (given by a translation permuting the hyperplanes in a "pencil", and $AGL_{d-1}(F)$ acting naturally in the hyperplanes). Then I would like to argue that the kernel of this mapping is 2-transitive. I haven't succeeded yet at finding a morphism or showing that the kernel should be 2-transitive.
Perhaps there is another approach, the group $ASL_d(F)$ is 2-transitive, so $H$ could be a subgroup of $ASL_d(F)$. I also haven't succeeded at proving the claim with this approach. Any help would be appreciated.