Find an example of a group morphism

Question

Find an example of a group morphism $f : G_1 \to G_2$ such that $H_1 ◅ G_1$ and $f(H_1)$ is not normal in $G_2$.

How would you go about answering this question?

Answer 1

Take $G_1=S_3$, $G_2=S_4$, $H_1=A_3=\{(1),(123),(132)\}$ and $f$ the natural embedding of $S_3$ in $S_4$.

Answer 2

Take $G_1=\{e, (12)(34), (13)(24),(14)(23)\}$, the abelian subgroup of $A_4$ isomorphic to Klein's Vierergruppe. It is a normal subgroup of $A_4$ and of $S_4$.

However, its subgroup $\langle\,(12)(34)\,\rangle$ while being normal in $V$, is not normal in $A_4$, since its conjugates, besides itself, are $\langle\,(13)(24)\,\rangle$ and $\langle\,(14)(23)\,\rangle$.

Answer 3

An idea to construct many counterexamples: Take $H_1=G_1$ and $G_2 = H \rtimes G_1$. Then, there is a natural monomorphism $\varphi : G_1 \to G_2 = H \rtimes G_1$, and $\varphi(H_1)$ is a normal subgroup if and only if $H \rtimes G_1 = H \times G_1$ (see here).