Let $f:X\rightarrow Y$ be a continuous map of topological spaces and let $\mathcal{F}$ be a sheaf (say of abelian groups to fix the idea) on $Y$. Define the following rule on open sets of $X$: $$ V\mapsto f'\mathcal{F}(V):=\varinjlim_{U\supseteq f(V)}\mathcal{F}(U) $$ One may check that $f'\mathcal{F}$ gives a presheaf on $X$. Under some strong assumptions one may show that $f'\mathcal{F}$ is a sheaf on $X$. For example, if $Y$ is Hausdorff, locally compact and paracompact and $f$ is a closed embedding then $f'\mathcal{F}$ is actually a sheaf (this is not a trivial exercise, it took me a while to figure that out).

So here is my question:

Q I would like to have a couple of (non-artificial ) examples where the presheaf $f'\mathcal{F}$ fails to be a sheaf in order to have a feeling for the possible geometrical (and/or topological) obstructions.