How does isomorphism of schemes induce maps on cohomology groups

Question

Let $f:X \to X$ be an isomorphism of schemes. Let $\mathcal{F}$ be sheaf of abelian groups on $X$.

When does $f$ induces a map $H^i(X, \mathcal{F}) \to H^i(X, \mathcal{F})$?

I've seen similar statements at many places at least certain types of $\mathcal{F}$ but never seen a proper explanation of how such a map exists?