How to show that the Flux hommorphism is surjective?

Question

I'm reading about the flux hommorphism for volume-preserving diffeomorphisms and I have tried to show that it's surjective. Even bearing in mind the proof of the surjectivity in the case of symplectic form I'm still unable to show it.

If $(M, \Omega)$ is an oriented manifold, the flux hommorphism for volum form can be defined as $$\widetilde{S_\Omega}(\phi_t)=\left[\int_0^1\phi_s^{\ast}(\imath_{\dot{\phi_t}})\Omega dt\right].$$

In my reading I've found two possible ways for answering this:

1. construct a right inverse and conclude that the map surjective.

2. I tried to use Proposition 3.1.6 P. 61 of the book of A. Banyaga "The Structure of Classical Diffeomorphism Groups" which states that if the mapping $$X \mapsto \imath(X)\Omega$$ is onto then the flux is onto, where $X$ is a vector field on the manifold. And in particular this is true for symplectic form and volume form.

But sincerely I don't know how to proceed. Can someone help me with this?