Suppose $\mathfrak g$ is a finite dimensional Lie algebra over a field on characteristic zero and $G$ is a finite group of automorphisms of $\mathfrak g$.



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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.



Equivariant Levi subalgebras.
Yes. This results and its variations are contained in papers by E.J. Taft:


* Invariant Wedderburn factors, Illinois J. Math. 1 (1957), N4, 565-573 <http://projecteuclid.org/euclid.ijm/1255380679> .
* Invariant Levi factors, Michigan Math. J. 9 (1962), N1, 65-68 DOI:10.1307/mmj/1028998623
* Orthogonal conjugacies in associative and Lie algebras, Trans. AMS 113 (1964), No.1, 18-29 DOI:10.2307/1994088


