Removing the p-solvability hypothesis from the induction lemma

Determine whether the conclusion of Lemma~\ref{lem:p-solvable}, namely equality of the p-rationality levels of a p'-degree character of a p-solvable subgroup and its induced character when the subgroup has p'-index and the character has p-rationality level at least 2, remains valid without assuming that the subgroup is p-solvable.

Background

Lemma~\ref{lem:p-solvable} proves p-rationality preservation for ordinary induction under the assumptions that the subgroup is p-solvable, has p'-index in the ambient group, and the inducing character has p'-degree and p-rationality level at least 2. The final paragraph explicitly leaves unresolved whether p-solvability is necessary. A positive answer would extend the proof of the theorem from p-solvable finite reductive groups to all cases with an F-stable parabolic subgroup, thereby proving the corresponding instance of Conjecture A.