Abelian quotients and orbit sizes for subgroups of linear groups

Keller and Yang formulated two conjectures relating abelian quotients of subgroups of finite linear groups to largest orbit sizes. After counterexamples were found to the original conjectures, two revised versions were proposed. We prove both revised conjectures for every finite group $G$ satisfying $F(G)=F^*(G)$. We also extend a regular-orbit theorem for nilpotent subgroups to completely reducible modules of mixed characteristic under the same hypothesis.