Say I know that $G_1,G_2$ are isomorphic groups, both with normal subgroups $N_1,N_2$ respectively. Can I take an isomorphism $psi$ between $G_1,G_2$ s.t $psi(N_1)=N_2$? Meaning that $psi$ also defines a isomorphism between $N_1$ and $N_2$?
The thm I’m trying to prove is if $G_1 cong G2, N1cong N2,N1unlhd G1, N2 unlhd G2$ then $G1/N1 cong G2/N2$, and proving the above statement I would be able to use the first isomorphism theorem and that’s it. Is it even true?
Any hint would be helpful on how to approach this problem.