Generalized reflection groups II

Let (G,X) be a pair consisting of a group G and a generating set X of G. Let K be a commutative ring and let M be a KG-module. In this work, I try to answer the following question: “Is it possible to find a K-algebra L such that M is a LG-module and such that the elements of X act as reflections of M?” If X is a set of involutions and if a presentation of G is known, then I can answer the above question.