About quotient groups with isomorphic factors

Question

In one of my previous questions (which is now deleted), I used, without proof, the "wrong" fact that if $A\cong B$ and $C\cong D$, and if $C$ is normal in $A$ and $D$ is normal in $B$, $A/C\cong B/D$.

Someone commented that it is not even true for a finite group, and I am wondering what a counterexample might look like. Also, under what circumstances (if any) is the above assertion valid? If I assert that it is finite and abelian, is it enough?

Thanks bunch in advance!

Answer 1

Set $A=B=\mathbb{Z}$, $C=2\mathbb{Z}$, $D=3\mathbb{Z}$.

It's also not true for finite and abelian groups: take $A=B=\mathbb{Z}/4\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$ and consider $C=2\mathbb{Z}/4\mathbb{Z} \times \{0\}$, $D=\{0\}\times \mathbb{Z}/2\mathbb{Z}$.

Answer 2

It is not true. It would be true if for an isomorphism $\phi\colon A$ to $C$ we had $\phi(B)= D$.