  We consider components of Hurwitz moduli space of G-Galois covers and set up
a powerful algebraic framework to study the set of corresponding equivalence
classes of monodromy maps. Within that we study geometric stabilisation by
various G-covers branched over the disc. Our results addresses the problem to
decide equivalence and stable equivalence algebraically. We recover a
homological invariant, which we show to distinguish the equivalence classes of
given boundary monodromy and Nielsen type, if the latter is sufficiently large
in the appropriate sense.
