Is the converse of the first isomorphism theorem true?

Question

Let $G,H$ be groups and $N$ be a normal subgroup of $H$.

Assume $G\cong H/N$.

Then, does there exist a epimorphism $\phi:H\rightarrow G$ with a kernel $N$?

Answer 1

Let $\pi : H \to H / N$ be the canonical projection. Let $\psi : H/N \to G$ be an isomorphism. Consider $\phi = \psi \circ \pi$.