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

Let $p>3$ be a prime. For each maximal subgroup $H⩽\mathrm{G}\mathrm{L}(d,p)$ with $\mathrm{\mid }H\mathrm{\mid }⩾p^{3d+1}$, we construct a $d$-generator finite $p$-group $G$ with the property that $\mathrm{A}\mathrm{u}\mathrm{t}(G)$ induces $H$ on the Frattini quotient $G\mathrm{/}\mathrm{\Phi }(G)$ and $\mathrm{\mid }G\mathrm{\mid }⩽p^{\frac{d^4}{2}}$. A significant feature of this construction is that $\mathrm{\mid }G\mathrm{\mid }$ is very small compared to $\mathrm{\mid }H\mathrm{\mid }$, 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 $G\mathrm{/}\mathrm{\Phi }(G)$, the construction yields groups with smallest nilpotency class, and in most cases, the smallest order.