Given an injective envelope E of a left R-module M, there is an associative Galois group Gal$({phi})$. Let R be a left noetherian ring and E be an injective envelope of M, then there is an injective envelope $E[x^{-1}]$ of an inverse polynomial module $M[x^{-1}]$ as a left R[x]-module and we can define an associative Galois group Gal$({phi}[x^{-1}])$. In this paper we describe the relations between Gal$({phi})$ and Gal$({phi}[x^{-1}])$. Then we extend the Galois group of inverse polynomial module and can get Gal$({phi}[x^{-s}])$, where S is a submonoid of $mathbb{N}$ (the set of all natural numbers).