A crossed homomorphism for groups acting on the circle

We construct a crossed homomorphism by using a group action on the circle and the Poincaré translation number. We relate it to the Euler class of the action in terms of the Hochschild–Serre spectral sequence. As an application, we answer a question of Calegari and Chen, which is on an explicit form of a certain crossed homomorphism on the mapping class group of the sphere minus a Cantor set.