The radical of the commutator subgroup Is the solvable radical of the commutator subgroup $G'$ of a group $G$ normal in $G$?

Yes, because the commutator subgroup is a characteristic subgroup of $G$. Then the claim follows from this duplicate:

Prove that if $H$ is a characteristic subgroup of $K$, and $K$ is a normal subgroup of $G$, then $H$ is a normal subgroup of $G$