Acyclic but not flasque sheaf of abelian group?

Question

Is there a sheaf of abelian groups which is acyclic but not flasque?

Maybe we can try $0\to \mathcal{F'}\to \mathcal{F}\to \mathcal{F''}\to 0$ where $\mathcal{F',F''}$ are flasque but $\mathcal{F}$ is not.

Answer 1

Consider $\mathbb{R}$ and constant sheaf on it. Cohomology of constant sheaf are just topological cohomology. So this sheaf has no higher cohomology ($\mathbb{R}$ is contractible).