# Computing maps in Leray spectral sequence

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?