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Let $f: X' \to X$ be a finite flat morphism of (nice) schemes and $\mathscr{A}/X'$ be a smooth commutative group scheme such that the Weil restriction $f_*\mathscr{A}/X$ is representable by a commutative group scheme. Are there criteria when $\mathrm{R}^1f_{*,\mathrm{fppf}}\mathscr{A}$ is representable by a (smooth) group scheme (or algebraic space)? For $\mathscr{A} = \mathbf{G}_m$, this is the Picard functor.

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