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 $H < G$, either $H'$ is core-free or 
$H' \leq \Phi(G)$, then is it true that $\Phi(G)$ necessarily is trivial?