The F-A characterization conjecture for finitely generated groups

Let $G$ be a finitely generated group, and let $G'$ denote its commutator subgroup. The group $G$ is called F-A when it can be expressed as the union of all its proper, normal, finite-index subgroups.

F-A characterization conjecture. $G$ is F-A if and only if

$$G/G'\text{ is non-cyclic}.$$

The paper presents this as one of two equivalent conjectures, motivated by characterization results for several classes of finitely generated groups, including simple, free, abelian, solvable, finite, and certain two-generator groups. Its general status is not resolved in the supplied text.