Endotrivial modules for the general linear group in a nondefining characteristic

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