First off, if , then , so . So , if it exists, is a subset of . So, it definitely feels like is the right choice. As for the rest, there is an isomorphism theorem for modules that is much like the ring and group isomorphism theorems.

It basically says that . Since you have that . Now, you are right that does not reside within so maybe you can try to find a submodule of that is isomorphic to and then show that . From there it'll follow that is indeed an isomorphism.