This is from the beginning of the section on group cohomology in Corps Locaux (English Edition).
Serre states that $A$ is an induced $G$-module if
(1) $A\cong A\otimes_\mathbb{Z}X$ for an abelian group $X$,
or, equivalently,
(2) $A=\bigoplus_{s\in G}s\cdot X$.
Is (1) a typo? This is a very strict condition not just on $A$, but also on $X$. It seems to me that the correct version of the first definition above should read $A\cong A\otimes_{\mathbb{Z}[X]}X$, which is in line with the usual corresponding notion for group representatinos over fields, but perhaps I'm just missing something simple.