For a finite group $G$, define $ R(G)=\cap\{K \triangleleft G \; | \; G/K \text{ is solvable}\}$ If $\alpha:G\rightarrow G_1$ is a group homomorphism, show that $\alpha[R(G)]\subseteq R(G_1)$.
This is part $(c)$ of the question - I've already proved that $R$ is the smallest normal subgroup of $G$ such that $G/R$ is solvable, so maybe that would come in useful. Thanks in advance!