A novel computational methodology under the umbrella of the so-called GS4-1 framework is presented in this series of articles for use in solving time-dependent heat transfer problems. In Part 1 of this series, we presented the framework for linear applications. In this Part 2, we show how this framework can be properly extended for use in nonlinear applications. The present computational framework and the underlying numerical methodology naturally inherit desirable features including second-order time accuracy, unconditional stability, and controllable numerical dissipation. Of noteworthy mention and significance is the fact that the framework, additionally, inherits new features that permit selective control of high-frequency damping for both the primary variable and its time derivative. Consequently, the development enables not only the analysis of long-term system dynamics to be satisfactory, but also readily enables one to capture the underlying physics. The current state of the art does not permit such features and consequently leads to physically incorrect dynamics of the system. Additionally, we describe in this article all the possible approaches that can be employed in the computational procedure to take into account the nonlinearity of the problem. The essential concepts and the significance of the present development are demonstrated through application to a simple radiation heat transfer problem for which an analytic solution is available.