On the lattice of regular transitive subgroup functors

We study the structure of the lattice $\mathrm{Reg}_\mathrm{tr}(\mathfrak X)$ of all regular transitive subgroup $\mathfrak X$-functors. We describe all hereditary formations $\mathfrak X$ for which the width of $\mathrm{Reg}_\mathrm{tr}(\mathfrak X)$ is finite and does not exceed $|\pi(\mathfrak X)|$, where $\pi(\mathfrak X)$ is the set of all prime divisors of the orders of the groups in $\mathfrak X$.