So I could barely understand the problem statement ("oriented boundary given by a surface $F$ having the map $f$"), nor how to proceed. Can I get some hints? Thank you.
Consider the smooth map $f: F \to S^2$ of degree $n$. For which values of $n$ does there exist a compact, oriented 3-manifold $X$ with oriented boundary given by a surface $F$ having the map $f$, and a closed 2-form $\eta \in \Omega^2(X)$ whose restriction to $F$ is $f^*\omega$? Prove that your answer is complete.