6.56 (1978)

Let $G = F \cdot \langle a \rangle$ be a Frobenius group with the complement $\langle a \rangle$ of prime order.
$\qquad$ a) Is $G$ locally finite if it is binary finite?
$\qquad$ b) Is the kernel $F$ locally finite if the groups $\langle a, a^g \rangle$ are finite for all $g \in G$?