Assume that $X$ is a vector field on a $n$ dimensional manifold $M$.Let $0\leq i,j \leq n$.

1.Is there a linear isomorphism $T:\Omega^i(M) \to \Omega^j(M)$ with $L_X T=T L_X$?

2.Is there a linear isomorphism $T:\Omega^i(M)/Z^i(M) \to \Omega^j(M)/Z^j(M)$ with $TL_X=L_XT$?

By $Z^i(M)$ we mean the space of closed differential $i$_form.

For a motivation please see the following post: