Strongly base-two groups

Let $G$ be a finite group, let $H$ be a core-free subgroup and let $b(G,H)$ denote the base size for the action of $G$ on $G\mathrm{/}H$. Let $\alpha (G)$ be the number of conjugacy classes of core-free subgroups $H$ of $G$ with $b(G,H)⩾3$. We say that $G$ is a strongly base-two group if $\alpha (G)⩽1$, which means that almost every faithful transitive permutation representation of $G$ has base size $2$. In this paper we study the strongly base-two finite groups with trivial Frattini subgroup.