We define a Flasque Sheaf on a site as one whose first Čech Cohomology vanishes for every covering of every object of the site. I know this definition is non-standard, but Lei Fu's 'Étale Cohomology Theory' uses this definition and shows it to be equivalent to the standard one (i.e, that all higher cohomology groups vanish for every object).
In the proof, he uses the following Lemma which I could not prove: If $$0\rightarrow F'\rightarrow F\rightarrow F''\rightarrow 0$$ is an exact sequence and $F'$ and $F$ are Flasque, so is $F''$. I am not entirely sure if this is true or if Fu has made a mistake, because I have never seen this claim made anywhere else.
Here is the relevant section in the book:
