Suppose $G$ is solvable, $N \vartriangleleft G$.
Let $f \in Hom(G,H)$. We have a normal series $\{e\}=G_0 \vartriangleleft G_1 \vartriangleleft ... \vartriangleleft G_n = G$ with $G_{i+1}/G_i$ abelian. Let $H_i = f(G_i)$. We denote $f_{i+1}(G)$ as the composition of $f$ and the quotient map $q_{i+1}(H_{i+1}) = H_{i+1}/H_i$, i.e. $f_{i+1} = f\circ q_{i+1}$.
Show $G_{i+1}/Ker(f_{i+1}) \simeq H_{i+1}/H_i$.
So, I know that $f_{i+1}$ is surjective. How do I get injectivity?