$\hat{G}$-invariant quasimorphisms and symplectic geometry of surfaces

Let $\hat{G}$ be a group and $G$ its normal subgroup. In this paper, we study $\hat{G}$-invariant quasimorphisms on $G$ which appear in symplectic geometry and 2-dimensional topology. As its application, we prove the non-existence of a section of the flux homomorphism on closed surfaces with higher genus. We also prove that Py's Calabi quasimorphism and Entov--Polterovich's partial Calabi quasimorphism cannot be extended to the group of symplectomorphism as partial quasimorphisms.