Let $f : X \to Y$ be an fpqc morphism of schemes, and let $\mathcal{G}$ be an $\mathcal{O}_{Y}$-module such that $f^{\ast}\mathcal{G}$ is quasi-coherent. Is $\mathcal{G}$ necessarily quasi-coherent?

I'd be happy to see any answers to the above question with "fpqc" replaced by "fppf" or "etale" as well.

Let $f : X \to Y$ be an fpqc morphism of schemes, and let $\mathcal{G}$ be an $\mathcal{O}_{Y}$-module (on the small Zariski site) such that $f^{\ast}\mathcal{G}$ is quasi-coherent. Is $\mathcal{G}$ necessarily quasi-coherent?

I'd be happy to see any answers to the above question with "fpqc" replaced by "fppf" or "etale" as well.