$H,K$ be subgroups of a group $G$ such that $HK$ is also a subgroup ; when is it possible that $HK$ is a homomorphic image of $H$?

Question

Let $H,K$ be subgroups of a group $G$ such that $HK$ is also a subgroup ; when is it possible that there exist a surjective group homomorphism from $H$ onto $HK$ ? If both $H,K$ are finite then it is easy to see that $H=HK$ i.e. $K \subseteq H$ ; other than that I have not been able to make any progress , Please help . Thanks in advance