I was trying to solve the following problem:
Let $G,K,H$ groups and $\text{φ}:G \rightarrow K, \text{ψ}: K \rightarrow H$ homomorfisms and $ker{\text{ψ}} \subseteq \text{φ}(G)$ . If $G,H$ are solvable, is K solvable?
I know generally that the class of solvable groups is closed to extensions. But how about that? Any hints?