Let $G$ be a group and $H, H'$ be subgroups of $G$ where $H$ is normal. Under which circumstances is $H \cap H'$ a normal subgroup of $H$?

Question

To be more specific, what kind of assumptions do I have to make for $H'$ to obtain this assertion? Which one are necessary and which one are sufficient?

For instance, we could say $H \subset H'$, but this case seems rather boring. Do you have some better ideas? Thank you!