Personally I would try to persuade someone who does have access to the paper to e-mail it to me, but that might not be legal, so I shouldn't have said it.
I have not thought this out in detail, but I think the following approach will work.
Incomplete argument deletedUse induction on $|G|$. Let $N$ be a minimal normal subgroup of $G$, so $N$ is an elementary abelian $p$-group for some prime $p$. Apply inductive hypothesis to $G/N$ to get $HN$ and $H^gN$ conjugate by an element of $\langle H,H^g \rangle$. So now we can assume that $HN=H^gN$.
Then $g \in N_G(HN)$ and so the Frattini property implies that $H$ and $H^g$ are conjugate in $HN$, and hence we can assume that $g \in HN$ and $HN=G$.
So $H$ acts irreducibly on $N$, and hence either $G=H$ and we are done, or $H$ is a complement of $N$ in $G$ and a maximal subgroup of $G$. Then either $H=H^g$ or $\langle H,H^g \rangle=G$ and we are done.