Yes, we take the map x+n |--------> (x+n, cl(x)) where x in G-N and n in N.

Abdelmajid Khadari, G/N isn't the set subtraction. It is quotient group - set of all cosets of N in G. I guess you missed that. Please, explain your point of view in more details if I am wrong. Thank you!

Peter T Breuer, great idea about composition! Thank you! But that is only one side of the problem. I mean that maybe that is not the only case. I found this information on the internet:

"Sometimes, but not necessarily, a group G can be reconstructed from G / N and N, as a direct product or semidirect product. The problem of determining when this is the case is known as the extension problem. An example where it is not possible is as follows. Z4 / { 0, 2 } is isomorphic to Z2, and { 0, 2 } also, but the only semidirect product is the direct product, because Z2 has only the trivial automorphism. Therefore Z4, which is different from Z2 × Z2, cannot be reconstructed."

So, I guess I need to look for the solution for that if there is any :) Interesting what is the current state of that problem. If somebody knows - please give some info and leave comment here.

Splitting problem is an important problem in group theory. The answer of your question can be a paper!!

As a simple example, consider the subgroup Z (integer numbers)  of Q (rationales numbers). Since Q is divisible and Z reduced, so Q is not isomorphic to Z product Q/Z.

In the category of abelian groups, if N is a divisible subgroup of G, then the answer of your question is yes.