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Let $f:X_{s1} \to X_{s2}$ be morphism of sites ( Here $X$ is some scheme $X_{s1}$ refers to the site on $X$).

Now using the Leray spectral sequence one gets the following exact sequence

$0 \to H^1(X_{s2}, f_{*}\mathcal{F} ) \to H^1(X_{S1}, \mathcal{F}) \to H^{0}(X_{s2},R^{1}f_{*}\mathcal{F}) \to H^2(X_{s2}, f_{*}\mathcal{F})$.

Is there any example where the maps of the above sequence has been explicitly computed? Say for example suppose I want to compute the map $H^{0}(X_{s2},R^{1}f_{*}\mathcal{F}) \to H^2(X_{s2}, f_{*}\mathcal{F})$ explicitly. Is there any example where I can find such a computation?

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