Surjective morphism of sheaves of sets

Question

If $\mathcal{F}, \mathcal{G}$ are sheaves of groups on a topological space $X$, then a surjection of sheaves $\mathcal{F}\rightarrow \mathcal{G}\rightarrow 0$ is a surjection of presheaves if the kernel $$\text{ker}(\mathcal{F}\rightarrow \mathcal{G})$$ is acyclic. My question is if we can adapt this to morphism of sheaves of sets, not groups. The problem is that there is not (to me) an obvious candidate of the kernel. My best guess would be to look at some fiber products $$\mathcal{F}\times_{\mathcal{G}} \mathcal{C}$$ where $\mathcal{C}$ is a "special" subsheaf of $\mathcal{G}$ (imitating the case of sheaves of groups, where $\mathcal{G}$ would be the unit subsheaf). Does such an approach work for the correct choice of "special"? I hope that if we set "special" to be the sheaves induced by global sections $\mathcal{G}(X)$ we get the right result, but don't know how prove it without using homological algebra.