Suppose $\mathfrak g$ is a finite dimensional Lie algebra over a field on characteristic zero and $G$ is a finite group of automorphisms of $\mathfrak g$.
Does there necessarily exist a Levi subalgebra of $\mathfrak g$ which is $G$-invariant?
By Levi subalgebra I mean a semisimple complement of the solvable radical, as in the Levi-Malcev theorem. My field is $\mathbb Q$... but if needed I could probably deal with extensions of scalars.
