Examples for failure of cohomology presheaf to be a sheaf

Question

Let $f:X\to Y$ be a continuous map and $F$ a sheaf on $X$. Consider the cohomology presheaf on $Y$ defined by $V\mapsto \mathrm H(f^{-1}V;F)$. I am looking for some instructive examples to understand when:

1. The cohomology presheaf is not a sheaf.
2. The cohomology presheaf is is sheaf.

For the first question, I thought perhaps the sheaf condition might fail if there are several "local cocycles" which glue to a trivial cocycle, but I'm struggling with examples. At least in homology, it doesn't seem possible for local holes to exist without global ones.