Combinatorial aspects of the Torelli-Johnson-Morita theory of surface automorphisms are extended to certain subgroups of the mapping class groups. These subgroups are defined relative to a specified homomorphism from the fundamental group of the surface onto an arbitrary group $K$. For $K$ abelian, there is a combinatorial theory akin to the classical case, for example, providing an explicit cocycle representing the first Johnson homomophism with target $\Lambda ^3 K$. Furthermore, the Earle class with coefficients in $K$ is represented by an explicit cocyle.
Geometriae Dedicata, 2013, Vol 167, Issue 1, p. 151-166