Let $\varphi: G \longrightarrow G'$ be a homomorphism group. If $N$ is a normal subgroup of $G$ then not necessarily $\varphi(N)$ is a normal subgroup of $G'$.
Of course the Ker $\varphi$ and Im $\varphi$ are subgroups, but don't see why if $N$ is a normal subgroup of $G$ then not necessarily $\varphi(N)$ is too. Any example?
Let $H$ be any subgroup of a group $G'$ which is not normal.
Pick $N=G=H$ and $\phi : G \to G'$ to be the inclusion.
P.S. In the case $\phi$ is onto, then every $y\in G'$ can be written as $y=\phi(x)$ and you get $y\phi(N)y^{-1}=\phi(xNx^{-1})=\phi(N)$. The problem in general is that $\phi(G)$ could be smaller than $G'$, which lead to the above example.