  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 $\alpha(G)$ be the
number of conjugacy classes of core-free subgroups $H$ of $G$ with $b(G,H)
\geqslant 3$. We say that $G$ is a strongly base-two group if $\alpha(G)
\leqslant 1$, 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.
