Invariant subspaces of an $F_2$-representation of the affine linear group of dimension 1

Question

Let $p$ be an odd prime (large if it matters) and let $G= Aff(\mathbb{F}_{p^2}) \cong \mathbb{F}_{p^2} \rtimes \mathbb{F}_{p^2}^*$ be the affine linear group acting on $\mathbb{F}_{p^2}$ by $x\mapsto ax+b$, $a\in \mathbb{F}_{p^2}^*$ and $b\in\mathbb{F}_{p^2}$. Consider the $\mathbb{F}_2$-representation $V=\{f\colon\mathbb{F}_{p^2}\to \mathbb{F}_2\}$ of $G$ with the action $f(x)\mapsto f(ax+b)$.

There are 4 invariant subspaces that I can easily find, namely the two trivial ones, the constant functions, and the space of $f$ with $\sum_{x} f(x)=0$.

Are there any more invariant subspaces?

Answer 1

There are no other invariant subspaces. As a consequence of Klemm's Satz 8(b) in here, we obtain the following: Let $G$ be a finite sharply $2$-transitive permutation group on $n\ge2$ letters. Then the permutation module for this action over a field of characteristic not dividing $n$ has only the four obvious submodules.

The group $G$ in the question is sharply $2$-transitive, and the given representation is the permutation module.