Given a reductive group $G$ over a finite field and a parabolic subgroup $P$ , I wonder what are the orbits in $G$ under the adjoint action of $P$. This should be standard, but I can only find results on the orbits inside $P$.

Actually, I would like to know which (say) permutation representations of the parabolic $Pto operatorname{Sym}(Omega)$ admit an extension to a $P$-equivariant map of sets $Gto operatorname{Sym}(Omega)$. But I do not know in which context this question falls.

I would already be very happy for an answer in the case $G=operatorname{GL}_3(mathbb{F}_p)$ and the $(2,1)$– parabolic $P=(operatorname{GL}_2(mathbb{F}_p)times operatorname{GL}_2(mathbb{F}_p))ltimes mathbb{F}_p^2$. I want to play around with a group-theoretic version of connection and parallel transport on the Grassmanian $operatorname{Gr}(3,2)$ over $mathbb{F}_p$.

Thanks and greetings,

Simon