Index of subgroups in a finite solvable group, with trivial Frattini subgroup (Exercise 3B.12 from Finite Group Theory, by M. Isaacs)

Yes, I regret that Problem 3B.12 of my group theory book is wrong. It should be replaced by the following:

Let $H \subseteq M \subseteq G$, where $M$ is a maximal subgroup of a solvable group $G$, and assume that the core of $M$ in $G$ is trivial. Show that $G$ has a subgroup with index equal to $|M:H|$.

I. M. Isaacs

$G=\mathrm{Alt}(4)$, $M=C_2^2$, $H=C_2$ is a counterexample, as $\mathrm{Alt}(4)$ doesn't have a subgroup of order $6$. This seems like a mistake in the book.