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We construct invariant quasimorphisms for groups acting on the circle.Furthermore, we provide a criterion for the non-extendablity of the resultingquasimorphisms and an explicit formula which relates the values of ourquasimorphisms to those of the Poincar\'{e} translation number. By using them,we show that the stable commutator length $\mathrm{scl}_G$ and the stable mixedcommutator length $\mathrm{scl}_{G,N}$ are not bi-Lipschitzly equivalent forthe surface group $G=\pi_1(\Sigma_{\ell})$ of genus at least $2$ and itscommutator subgroup $N = [\pi_1(\Sigma_{\ell}), \pi_1(\Sigma_{\ell})]$. We alsoshow the non-equivalence for a pair $(G,N)$ such that $G$ is the fundamentalgroup of a $3$-dimensional closed hyperbolic mapping torus. These pairs serveas the first family of examples of such $(G,N)$ in which $G$ is finitelygenerated.

Invariant quasimorphisms for groups acting on the circle and non-equivalence of SCL