Are quotient groups isomorphic if base groups and normal subgroups are?

[zcdfhn]

Prove or disprove:

Suppose N is a normal subgp of G and N' is a normal subgp of G'. If G is isomorphic to G' and N is isomorphic to N' does that mean that G/N is isomorphic to G'/N'?

I was trying to work out a proof until my professor told us to think of subgroups of the integers when doing this problem. So now I'm trying to disprove it through a counterexample. I have been stuck on this question for a while and I would appreciate any help.

Thank you.

 

[dx]

Hint: { ... -4, -2, 0, 2, 4 ... } is isomorphic to { ... -2, -1, 0, 1, 2, ... }.