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Let $X$ be a scheme, $F$ and $G$ be sheaves in groups on $X$ for the étale topology and $f:F\rightarrow G$ a morphism of group sheaves. Assume that there exists an étale covering $i:U\rightarrow X$ such that the pullback $i^{*}f:i^{*}F\rightarrow i^{*}G$ admits a section, can I lift this section to a section of $f:F\rightarrow G$?

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