Borho in this paper showed that a sheet (i.e. a suitable union of adjoint orbits of the same dimension) in a complex semisimple Lie algebra $\mathfrak{g}$ is classified by a pair $(\mathfrak{l},O)$ where $\mathfrak{l}$ is a Levi subalgebra of $\mathfrak{g}$ and $O$ is a rigid nilpotent orbit of $\mathfrak{l}$.

How does the story go in the case of a sheet in the corresponding group $G$? I guess I need to replace the Levi algebra with a pseudo-Levi group. I'd appreciate if you could suggest a reference to look at.