$\DeclareMathOperator\SL{SL}$Let $K$ be a field and let $G$ be a $K$-defined, closed, algebraic subgroup of $\SL_2$. Denote by $\mathcal{C}$ the site whose objects are $K$-schemes, with your favorite $G$-topology, say Zariski/fppf/w.e., and consider the map of presheaves on $\mathcal{C}$:

$$\pi: \SL_2\longrightarrow G\setminus \SL_2,$$ i.e., $\pi$ sends points of $\SL_2$ into the set of their $G$-orbits.

I would like to know under which conditions on $G$, is it guaranteed that there exists a section to $\pi$ (as a natural transformation of presheaves).