Let f: X {long rightwards arrow} X be a self-map of a topological space. In 1992, M. Woo and J. Kim introduced the q-Nielsen theory which is a tool to estimate the Nielsen number. In this paper, we compare the q-Nielsen theory to the mod H Nielsen theory. We develop an algebraic formulation of the mod H theory which does not require the condition f(H) ⊂ H based on the universal covering space instead of the regular cover, as well as an algebraic formulation of the q-Nielsen theory. In fact, we show that the q-Nielsen theory is the same as the mod H theory for H = ker(qπ: π1(X) {long rightwards arrow} π1(Y)), where qπ is the induced homomorphism of q and Y is a topological space. We also provide applications of the q-Nielsen theory and the mod H theory in the case where f(H) is not contained in H, as well as applications for wedge product spaces of nonaspherical types. © 2012 Springer Basel AG.