Suppose that $G$ is a finite minimal non-solvable group (i.e. a non-solvable group
all of whose proper subgroups are solvable) whose Frattini subgroup $\Phi(G)$
is abelian. If for every subgroup $H < G$, $H'$ is core-free, then is it true that $G$ is simple (or equivalently $\Phi(G)$ is trivial)?