In a previous work, we have investigated an automata-theoretic property of numeration systems associated with quadratic Pisot units that yields, for every such number `, a certain group G ` . In this paper, we characterize a cross-section of a congruence fl ` of Z 4 that had arisen when constructing G ` . This allows us to completely describe the quotient H ` of Z 4 by fl ` , that becomes then a second group associated with `. Moreover, the cross-section thus described is strikingly similar to the symbolic dynamical system associated, by a theorem of Parry, with the two numeration systems attached to `. The proof is combinatorial, and based upon rewriting techniques. R'esum'e Dans un article pr'ec'edent, nous avions associ'e `a chaque nombre de Pisot quadratique unitaire ` un certain groupe G ` par le biais de la construction d'un automate qui r'ealise le passage entre les repr'esentations des entiers dans deux syst`emes de num'eration naturellement attach'es `a `. Dans cet arti..
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