Suppose $ F : M \to M$ is diffeomorphism of smooth manifold $M$, and suppose $F^* \nu = \nu$ for differential 2-form $\nu$. Let $p: \tilde{M} \to M$ denote universal cover of $M$, and suppose $p^*\nu$ is exact differential 2-form on $\tilde{M}$.
Is it always possible to choose differential 1-form $\mu$ on $\tilde{M}$ such that $d \mu = p^* \nu$ and lift $\tilde{F} : \tilde{M} \to \tilde{M}$ of $F$ (so $\tilde{F}$ is diffeomorphism such that $p \circ \tilde{F} = F \circ p$) such that $\tilde{F}^*\mu = \mu$?
Idea is start with some differential 1-form $\mu'$ such that $p^* \nu = d\mu'$ and then average it somehow to make invariant.