In a previous question, I had two non-isomorphic groups, but which had isomorphic subgroups and factor groups. So I had groups $G_1$ and $G_2$ with normal subgroups $N_1$ and $N_2$ such that $N_1\cong N_2$ and $G_1/N_1\cong G_2/N_2$, but $G_1\not\cong G_2$.
This is seems to mean that, unlike normal factoring, a group can’t be reconstructed (up to an isomorphism) from its factors.
Is that right? Is there a similar notion that does allow reconstruction?