Repeated Richardson Extrapolation can successfully be used in the efforts to improve the efficiency of the numerical treatment of systems of ordinary differential equations (ODEs) mainly by increasing the accuracy of the computed results. It is assumed in this paper that Implicit Runge-Kutta Methods (IRKMs) are used in the numerical solution of systems of ODEs. If the order of accuracy of the selected IRKM is p, then the order of accuracy of its combination with the Repeated Richardson Extrapolation is at least p+2 (assuming here that the right-hand-side of the system of ODEs is sufficiently many times continuously differentiable). However, it is additionally necessary to establish that the absolute stability properties of the new numerical methods (that are combinations of the Repeated Richardson Extrapolation and the selected IRKMs) are preserved, and this is an extremely difficult problem. Results related to the stability of the computations are derived and numerical tests with a two-parameter system of three ODEs and an atmospheric chemical scheme with 56 compounds, which is defined mathematically by a very stiff and ill-conditioned system of non-linear ODEs, are presented. The research results described in this paper can be considered as a continuation of the study carried out in Zlatev et al.: Richardson Extrapolation: Practical Aspects and Applications. De Gruyter, Berlin (2017).