If $G$ is a reductive algebraic group over an algebraically closed field of positive characteristic $p$, and $G$ is defined over the prime field, we have the Frobenius morphism $F: G\to G$, which for each positive number $r$ leads to two subgroups of $G$, namely $G_r$ which is the kernel of the $r$'th iteration of $F$, and $G^{F^r}$ which is the fixed points of the same.
Now, $G^{F^r}$ normalizes $G_r$ (since $G_r$ is normal in $G$), and they clearly intersect trivially. So in $G$ the subgroup $G^{F^r}G_r$ is a semidirect product.
My question is whether this subgroup and its representations has been studied. We certainly know some of its irreducible representations, as any irreducible representation of $G_r$ or $G^{F^r}$ extend to $G$ (and they even have the same irreducibles when these are seen as $G$-modules). So one of the main questions would be whether there are any other irreducible representations of this group.
Of course, one could also consider $G^{F^r}G_{r'}$ with $r\neq r'$ and ask the same questions.
The reason I am interested in this group is that it seems like it might provide a more direct link between the representation theories of $G_r$ and $G^{F^r}$, which are certainly very similar.
It might also provide an additional stepping stone for comparing the representation theories of $G_r$ or $G^{F^r}$ with that of $G$, since one of the ways one often does such comparison is via the induction from either $G_r$ or $G^{F^r}$ to $G$, and both of these factor through this subgroup.
(I asked this question in the representation theory chat room, trying to get the discussion going, but I realized it was focused enough that it might as well be an actual question here).
