We know if a compact Lie group $G$ acts smoothly on a smooth manifold $M$, then for each $k$-form $\omega$ on $M$, we can simply construct a $G$-invariant $k$-form by "averaging over translations by $G$".

Now, suppose we have a (discrete) subgroup $\Gamma$ of ${\rm diff}(M)$, the full diffeomorphism group of $M$. For example, $\Gamma$ may be all of ${\rm diff}(M)$. Then there may not exist any $\Gamma$-invariant differential forms on $M$. But, is there any (principal) fiber bundle over $M$, with a natural lifting of the action of $\Gamma$, which carries $\Gamma$-invariant differential forms of a certain order $k$, or of any arbitrary order? I suspect the Jet bundle $J_k(M)$ of order $k$, or ultimately, the infinite jet bundle $J_\infty(M)$ may give the answer. But I am not sure.

The case $\Gamma$ equals integers, namely, generated by a single diffeomorphism is also quite interesting to me. Namely, given a single diffeomorphism $\phi$ on a manifold, can we construct a fiber bundle on $M$ that carries a $\bar\phi$-invariant differential form? Here, $\bar\phi$ is the ``natural" lifting of $\phi$ to the fiber bundle in question.