Let $G$ be a commutative algebraic group stack over $\mathbb{F}_q$ (I don't really care about the precise definition: I'm secretly thinking about the Picard stack of a projective curve). To what extent do the standard results on the Lang isogeny $x \mapsto \text{Fr}_q(x) \cdot x^{-1}$ carry over from the theory of group schemes over $\mathbb{F}_q$? For example, I don't see any reason for this map to be representable. But I would like to have a sense in which it is "etale with group $G(\mathbb{F}_q)$." Can this be done?

Let $G$ be a commutative algebraic group stack over $\mathbb{F}_q$ (I don't really care about the precise definition: I'm secretly thinking about the Picard stack of a projective curve). To what extent do the standard results on the Lang isogeny $x \mapsto \text{Fr}_q(x) \cdot x^{-1}$ carry over from the theory of group schemes over $\mathbb{F}_q$? For example, I don't see any reason for this map to be representable. But I would like to have a sense in which it is "etale with group $G(\mathbb{F}_q)$." Can this be done?

Edit: I think that for the Picard stack $\text{Pic}$ of a projective curve, the fiber of the Lang map $\text{Pic} \to \text{Pic}$ over the trivial line bundle is identified with $$\coprod_{\text{Pic}(X)} \bullet / \mathbb{F}_q^{\times}$$ (here $\bullet = \text{Spec } \mathbb{F}_q$ and the disjoint union is taken over the Picard group of isomorphism classes). So the Lang map indeed fails to be representable, but my intuition suggests that it is "smooth of relative dimension zero." Unfortunately, my technical facility with stacks is inadequate to formulate a statement along these lines.