Let Ĝ be a group and G its normal subgroup. In this paper, we study Ĝ-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.