Let $X \to Y$ be a degree 2 etale cover of smooth projective geometrically connected curves over some field. Consider the induced map on their Picard schemes. This map is never injective; in fact the covering classifies a nontrivial line bundle on $Y$ together with a trivialization of its square, and this bundle pulls back to the trivial bundle on $X$.
The question is: are there any more line bundles on $Y$ trivialized under pullback? Note that there are only finitely many because they are all forced to be 2-torsion.