This is a result of Artin and Tate found in Serge Lang's Algebra. (Page 166.)
Let $G$ be a finite group operating on a set $S$. For $w\in S$, denote $1\cdot w$ by $[w]$, so that we have the direct sum $ \mathbb{Z}\langle S\rangle=\sum_{w\in S}\mathbb{Z}[w]. $ Define an action of $G$ on $\mathbb{Z}\langle S\rangle$ by defining $\sigma[w]=[\sigma w]$, and extending $\sigma$ to $\mathbb{Z}\langle S\rangle$ by linearity. Let $M$ be a subgroup of $\mathbb{Z}\langle S\rangle$ of rank $\#(S)$. Show that $M$ has a $\mathbb{Z}$-basis $\{y_w\}_{w\in S}$ such that $\sigma y_w=y_{\sigma w}$ for all $w\in S$.
I know this is an adaptation of a result in Artin and Tate's notes on Class Field Theory. I found proofs of similar ideas in Lang's Algebraic Number Theory (Theorem 1 of page 190), and in Nancy Childress's book here.
I tried reading through the proofs and the relevant lemmas, but have had difficulty deciphering them and putting them in the context of Lang's statement. Is there perhaps a dumbed down proof of the above statement more suited to using knowledge of basic groups, rings, and modules? Thanks.