Projectivity of the Absolute Galois Groups of Multi-Valued Fields

In the article, we establish the projectivity property for the enriched Galois groups of multi-valued fields having a near Boolean family of valuation rings and satisfying the arithmetic local-global principle $\mathrm{LG}_A$. This property extends the corresponding property that was proved by the author for the case of Boolean families. As application, we give a sufficient condition for the inverse Galois problem to be solvable. This condition can be applied in particular to the wonderful extensions of the field of rational numbers.