Also in MO.
Let $\mathcal{F}$ be sheave over $S_\mathrm{fpqc}$. We say $\mathcal{F}$ is a fpqc-locally constant sheaf (of finitely generated abelian groups) if there exists a fpqc covering $(S_i\to S)_{i\in I}$ s.t. $\mathcal{F}|_{S_i}$ is a constant sheaf (associated with a finitely generated abelian group).
Clearly if $\mathcal{F}$ is étale-locally constant then $\mathcal{F}$ is fpqc-locally constant as every étale covering is a fpqc covering see tag 022C. Is the reverse ture?
I believe the reverse is true because a finite type group scheme over multiplicative type (fpqc-locally dual of constant group scheme associated with finitely generated abelian group) can be trivialised with a finite étale surjective map (stronger than trivializsed by an étale covering).
I think there should be a direct proof showing the fact that "fpqc-locally constant sheaf can be trivialised by an étale covering or even a finite étale surjective map" without mentioning group scheme. Also I'm not sure if the condition of requiring the associated constant set to be finitely generated abelian groups necessary.