On $\mathfrak{X}$-transitive groups and conjugate separable $\mathfrak{X}$-subgroups

For a given variety of groups $\text{X}$, we develop a systematic theory of $\text{CSX}$-groups and $\text{XT}$-groups, extending ideas proposed in \cite{Shah}. We analyze the interplay between these classes, describe their structural properties, and examine their connections with equational domains and residually $A$-free groups. Furthermore, we prove by elementary means that every finite $\text{CSX}$-group lies in $\text{X}$.