Bijection between $\mathcal{G}\subset \mathcal{L}\subset \mathcal{F}$ and $\mathcal{L}\subset \mathcal{F}/\mathcal{G}$

Question

Let $\mathcal{G}\subset \mathcal{F}$ be two sheaves (say of $A$-module with a suitable ring $A$) on a topological space $X$. Is there an increasing correspondance between the sheaves $\mathcal{L}$ such that $\mathcal{G}\subset \mathcal{L}\subset \mathcal{F}$ and the subsheaves of the sheaf $\mathcal{F}/\mathcal{G}$?

Many thanks!