Coordinate retractions and the image of the Magnus–Shmelkin embedding

Let \\(F=F(X)\\) be a free group, let \\(R\\trianglelefteq F\\), and let \\(\\mathcal V\\) be a variety of groups. We give a coordinate criterion for the image of the Magnus–Shmelkin embedding of \\(F/\\mathcal V(R)\\) in the corresponding verbal wreath product. For any normalized section of \\(F\\to F/R\\), we construct an idempotent endomorphism \\(\\rho_s\\) of the relatively free base. An element \\((b,q)\\) belongs to the embedded group precisely when \\(\\rho_s(b)=b\\tau_q^{-1}\\), where \\(\\tau_q\\) is the base coordinate of the chosen representative of \\(q\\). This gives a structural answer to Kourovka Problem 7.58 without restrictions on the rank, normal subgroup, or variety. We also prove the exact kernel of the canonical map by a section-loop evaluation.