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/H. Let α(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 α(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.