Endotrivial modules for the general linear group in a nondefining characteristic

Suppose that $G$ is a finite group such that $\SL(n,q)\subseteq G \subseteq \GL(n,q)$, and that $Z$ is a central subgroup of $G$. Let $T(G/Z)$ be the abelian group of equivalence classes of endotrivial $k(G/Z)$-modules, where $k$ is an algebraically closed field of characteristic~$p$ not dividing $q$. We show that the torsion free rank of $T(G/Z)$ is at most one, and we determine $T(G/Z)$ in the case that the Sylow $p$-subgroup of $G$ is abelian and nontrivial. The proofs for the torsion subgroup of $T(G/Z)$ use the theory of Young modules for $\GL(n,q)$ and a new method due to Balmer for computing the kernel of restrictions in the group of endotrivial modules