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We say that a group is anti-solvable if all of its composition factors arenon-abelian. We consider a particular family of anti-solvable finite groupscontaining the simple alternating groups for $n\neq 6$ and all 26 sporadicsimple groups. We prove that, if $K$ is a perfect field and $X$ is ahomogeneous space of a smooth algebraic $K$-group $G$ with finite geometricstabilizers lying in this family, then $X$ is dominated by a $G$-torsor. Inparticular, if $G=\mathrm{SL}_n$, all such homogeneous spaces have rationalpoints.

On homogeneous spaces with finite anti-solvable stabilizers