The abelianizations of Weyl groups of root systems extended by abelian groups

We investigate the class of root systems R obtained by extending an irreducible root system by a torsion-free group G. In this context there is a Weyl group and a group with the presentation by conjugation. We show under additional hypotheses that the kernel of the natural homomorphism is isomorphic to the kernel of , where and denote the abelianizations of and respectively.