Functoriality of sheaf Cohomology

Question

Let $f:Y\rightarrow X$ be a morphism of schemes. I would like to construct a natural map of homologies $$H^{q}(X,F)\rightarrow H^{q}(Y,f^{*}(F))$$ where $F$ is a sheaf on X. My idea is to take an injective resolution $I^{\bullet}$ of $F$ and $J^{\bullet}$ an injective resolution of $f^{*}(F)$. Of course the resolution $I^{\bullet}$ is flasque, so I think if I prove that $f^{*}(I^{\bullet})$ is flasque then I am done because there is a map $I^{\bullet}\rightarrow f_{*}f^{*} I^{\bullet}$. But I am stuck on how to do this. Any ideas?