Finite groups with Frobenius subgroup

Suppose the normalizer $N$ of a subgroup $A$ of a simple group $G$ is a Frobenius group with kernel $A$, and the intersection of $A$ with any other conjugate subgroup of $G$ is trivial, and suppose, if $A$ is elementary Abelian, that $|A|>2n+1$, where $n=|N:A|$. It is proved that if $A$ has a complement $B$ in $G$, then $G$ acts doubly transitively on the set of right cosets of $G$ modulo $B$, the subgroup $B$ is maximal in $G$, and $|B|$ is divisible by $|A|-1$. The proof makes essential use of the coherence of a certain set of irreducible characters of $N$.