3. Lifts

Base change
 $F$ : a $p$adic field of characteristic $0.$
 $G_F$ : an unramified linear group (quasisplit over $F$ and split over a finite unramified extension $L$ of $F.$)
 $\widetilde{G}_F$ : a covering group of degree coprime to $p.$
Problem 3.1.
[M. Weissman] Given a genuine irreducible representation $\widetilde{\pi}$ of $\widetilde{G}_F,$ describe $Lift_{L/F}(\widetilde{\pi})$ the base change lift of $\widetilde{\pi}.$ Note that $Lift_{L/F}(\widetilde{\pi})$ is supposed to be the virtual representation of $\widetilde{G}_L.$ (a) Define $\widetilde{G}_L.$
 (b) List axioms to characterize $Lift_{L/F}(\widetilde{\pi}),$ including explicit description of norm map.

This is a preliminary problem to Problem basechange
Problem 3.2.
Does $Gal(L/F)$ act on $\widetilde{G}_L$? 
Problem 3.3.
[E. Lapid] Make sense of transferring representation from $\widetilde{G}_F$ to $\widetilde{G'}_F,$ where two linear groups $G_F$ and $G'_F$ are inner forms each other. 
Problem 3.4.
[F. Adams] Is there a natural lifting from representations of $G_F$ to $\widetilde{G'}_F$?
Cite this as: AimPL: Automorphic forms on covering groups, available at http://aimpl.org/autoformcovergp.