Given a group G a quasimorphism on the group is a function f from the group to the reals which is almost a homomorphism i.e. |f(xy)-f(x)-f(y)| < C for all x, y in G and a universal constant C. I will talk about a work which is a small step in an attempt to find something in common to as many as possible of the known constructions of quasimorphisms. The idea is that quasimorphisms can be obtained via "local data" of group actions on certain appropriate spaces. In a rough manner, the principle says that instead of starting with a given group, and try to build or study its space of quasimorphisms, we should start with a space with a certain structure, in such a way that groups acting on this space and respecting this structure will automatically carry quasimorphisms. For countable groups it seems worthwhile chasing further this idea. Where for bigger groups, like structure preserving groups, meaningful examples are still under study and investigation, which I might say something about, according to time limitations.