The map you have defined does not seem to have anything to do with the Second Isomorphism Theorem that I know. Are you trying to prove this? Lattice theorem - Wikipedia, the free encyclopedia

If you think about your map for a moment, you realise that it simply regurgitates the contents of all of the cosets in ; therefore, the map does not depend on representatives at all.

I also think that your map is just a complicated way of defining , where is the canonical homomorphism.