# Isomorphism between groups and subgroups.

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.