I want to find an example of a homomorphism $f:G\to H$ such that $A$ is a normal subgroup of $G,$ but $f(A)$ is not so in $H.$
I know that if $f$ was onto $f(A )$ must be normal , but otherwise i want to find an example!
Take $A=G$ as a subgroup of $H$ that is not normal. Then define $f:G\rightarrow H$ by $x\mapsto x$
Notice here that $A=G$ is a normal subgroup of itself.
$f(A)=f(G)=G$ is not normal in $H$
Let $H < G$ which is not normal, and the injection $H \to G$. Then $A = H$ is normal in $H$, but not in $G$.