Product of a biprimary and a 2-decomposable group

Suppose a finite group $G$ is the product of a subgroups $A$ and $B$ of coprime orders, and suppose the order of $A$ is $p^aq^b$, where $p$ and $q$ are primes, and $B$ is a 2-decomposable group of even order. Assume that a Sylow $p$-subgroup $P$ is cyclic. If the order of $P$ is not equal to 3 or 7, then $G$ is solvable. If $G$ is nonsolvable and $G$ contains no nonidentity solvable invariant subgroups, then $G$ is isomorphic to $PSL(2, 7)$ or $PGL(2, 7)$.