Kodaira fibrations are topological surface-by-surface bundles with infinite monodromy which admit a complex structure. The first examples of Kodaira fibrations were constructed by Kodaira and by Atiyah in the 1960s. Since then the geometric and group theoretical properties of Kodaira fibrations have attracted much attention. In this talk we will explain how one can use Kodaira fibrations to construct a virtual embedding of the mapping class group of a once punctured surface of any genus into the mapping class group of a closed surface of higher genus. The homomorphisms we construct have the property that their images are irreducible. This is joint work with Pierre Py.