Given groups $ G_1 , G_2 $ and $H_1 \unlhd G_1 , H_2 \unlhd G_2$, show that $\varphi: G_1 \to G_2$ induces $\phi: \frac{G_1}{H_1} \to \frac{G_2}{H_2}$ if $\varphi (H_1) \subset H_2$.
So taking $\varphi: G_1 \to G_2 $ and $\pi : G_2 \to \frac{G_2}{H_2} $,
we have $\overline{\varphi} : G_1 \to \frac{G_2}{H_2} $ and $\overline{\varphi} = \pi \circ \varphi $.
Take $\psi : G_1 \to \frac{G_1}{H_1} $ by the factorization theorem $\exists \ \Phi : \frac{G_1}{H_1} \to \frac{G_2}{H_2} $.
I don't understand why this is not the end and I must consider $\varphi (H_1) \subset H_2$