$ABA$-groups with cyclic subgroup $B$

Some criteria to the solubility of groups of the form $G=ABA$ with a nilpotent subgroup $A$ and a cyclic subgroup $B$ are derived. In particular, it is proved (using the classification of the finite simple groups) that the finite group $G=ABA$ is soluble if $A$ is a nilpotent group of odd order and $B$ is a cyclic group and $(|A|,|B|)=1$.