Let $X\to B$ be a connected (un-ramified) covering map with group of deck transformations $G$; e.g. $\mathbb{C}\to T= \mathbb{C}/(\mathbb{Z}+i\mathbb{Z})$. The covering may be finite or infinite.

I am looking for a reference for the following statement (finite case is easy; thus, mainly for the infinite case):

Every $G$-invariant cohomology class on $X$ is pull-back of a cohomology class from the base.

RMK: I am not sure this is going to be true always; I just need it for quotients of maximal abelian covering.