Question about Quotient of Product of Subgroups

Question

Let $G$ be a group, $H, K, N$ be subgroups such that $N$ is normal in $G$ and $K$ is normal in $H$. I would like to know whether there exists an injective homomorphism from $\frac{NH}{NK}$ to $\frac HK$. So far I have shown that there is an injective homomorphism from $\frac {NH}{NK}$ to $\frac H{(N\cap H)K}$ given simply by $f(nh(NK)) =h (N\cap H)K$. But I could not proceed from there. I have considered the possibility of a counterexample, but I could not come up with one as well.