Quasimorphisms on groups provide a flexible tool to study bounded cohomology, stable commutator length and large-scale geometry. They have been studied for a long time within geometric group theory, symplectic geometry, and dynamical systems. In recent years, the extension problem of quasimorphism gains much attention, that is: when and how does a quasimorphism defined on a subgroup extend to a quasimorphism on the whole group?

In this talk, I will discuss some recent progress on this problem. I will show how to reprove a result of Hull and Osin with a much simpler proof and how to produce new examples of hierarchically hyperbolic quotients of mapping class groups, which answers a question of Fournier-Facio, Mangioni, and Sisto.