  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 low dimensional topology. As its application, we prove the non-existence of
a section of the flux homomorphism on closed surfaces of higher genus. We also
prove that Py's Calabi quasimorphism and Entov-Polterovich's partial Calabi
quasimorphism are non-extendable to the group of symplectomorphisms. We show
that Py's Calabi quasimorphism is the unique non-extendable quasimorphism to
some group.
