Groups saturated with finite Frobenius groups with complements of even order

We prove a theorem stating the following. Let $G$ be a periodic group saturated with finite Frobenius groups with complements of even order, and let $i$ be an involution of $G$. If, for some elements $a,b\in G$ with the condition $|a|\cdot|b|>4$, all subgroups $\langle a,b^g\rangle$, where $g\in G$, are finite, then $G=A\leftthreetimes C_G(i)$ is a Frobenius group with Abelian kernel $A$ and complement $C_G(i)$ whose elementary Abelian subgroups are all cyclic.