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 W and a group U with the presentation by conjugation. We show under additional hypotheses that the kernel of the natural homomorphism U→W is isomorphic to the kernel of Uab→Wab, where Uab and Wab denote the abelianizations of U and W respectively.