Solvability and nilpotency of expanded groups relative to the nearring of zero and congruence preserving functions

In this paper we study solvability and nilpotency of an expanded group relative to the nearring of zero and congruence preserving functions of the expanded group. A particular case of interest is when the expanded group is a nearring module. This leads to consequences when an expanded group is viewed as a nearring module over its nearring of zero preserving polynomial functions.