Maximal linear groups induced on the Frattini quotient of a $p$-group

Let $p>3$ be a prime. For each maximal subgroup $H \leqslant \mathrm{GL}(d,p)$ with $|H|=p^{\mathrm O(d^2)}$, we construct a $d$-generator finite $p$-group $G$ with the property that $\mathrm{Aut}(G)$ induces $H$ on the Frattini quotient $G/\Phi(G)$ and $|G|= p^{\mathrm O(d^4)}$. A significant feature of this construction is that $|G|$ is very small compared to $|H|$, shedding new light upon a celebrated result of Bryant and Kov\'acs. The groups $G$ that we exhibit have exponent $p$, and of all such groups $G$ with the desired action of $H$ on $G/\Phi(G)$, the construction yields groups with smallest nilpotency class, and in most cases, the smallest order.