The ambiguity index of an equipped finite group

In the paper (Kulikov in Sb Math 204(2):237–263, 2013), the ambiguity index (Formula presented.) was introduced for each equipped finite group (Formula presented.). It is equal to the number of connected components of a Hurwitz space parametrizing coverings of a projective line with Galois group (Formula presented.) assuming that all local monodromies belong to conjugacy classes (Formula presented.) in (Formula presented.) and the number of branch points is greater than some constant. We prove in this article that the ambiguity index can be identified with the size of a generalization of so called Bogomolov multiplier (Kunyavskiĭ in Cohomological and Geometric Approaches to Rationality Problems. Progress in Mathematics, vol 282, pp 209–217, 2010), see also (Bogomolov in Math USSR-Izv 30(3):455–485, 1988) and hence can be easily computed for many pairs (Formula presented.). In particular, the ambiguity indices are completely counted in the cases when (Formula presented.) are the symmetric or alternating groups.