  Oda's problem, which deals with the fixed field of the universal monodromy
representation of moduli spaces of curves and its independence with respect to
the topological data, is a central question of anabelian arithmetic geometry.
This paper emphasizes the stack nature of this problem by establishing the
independence of monodromy fields with respect to finer special loci data of
curves with symmetries, which we show provides a new proof of Oda's prediction.
