Let $B$ be a compact manifold, and $\hat{B}\to B$ be the maximal abelian covering of $B$; i.e. $\hat{B}$ is the quotient of the universal cover with respect to the commentator subgroup of $\pi_1(B)$. Given $H\subset H_1(B)$, we can further quotient $\hat{B}$ with respect to $H$ to get a covering $X\to B$ with group of deck transformation $G=H_1(B)/H$.

Is it true\false that every $G$-invaraint cohomology class on $X$ is a pull-back from $B$.