The emerging field of High Energy Atmospheric Physics studies events producing high energy particles and associated with thunderstorms, such as terrestrial gamma-ray flashes and gamma-ray glows. Understanding these phenomena requires appropriate models of the interaction of electrons, positrons and photons with air and electric fields. This work is made as a continuation of Rutjes et al. (2016), now including the effects of electric fields. We investigated results of three codes used in the community (Geant4, GRRR and REAM), for simulating the process of Relativistic Runaway Electron Avalanches. From analytical considerations, we show that the avalanche is mainly driven by electric fields and the ionisation and scattering processes determining the minimum energy of electrons that can runaway. To investigate this point further, we used a first simulation set-up to estimate the probability to produce a RREA from a relevant range of electron energies and electric field magnitudes. We found that the stepping methodology is important, and the stepping parameters have to be set up very carefully for Geant4. For example, a too large step size can lead to an avalanche probability reduced by a factor of 10, or a 40 % over-estimation of the average electron energy. Furthermore, the probability for the particles below 10 keV to accelerate and participate in the penetrating radiation is actually negligible for the full range of electric field we tested ( E < 3 MV/m). The added value of using models able to accurately track low energy particles (< 10 keV) is minor, and is mainly visible for high E -fields ( E > 2 MV/m). In a second simulation set-up, we compared the physical characteristics of the avalanches produced by the four models: avalanche (time and length) scales, time to self-similar state and photon/electron energy spectra. The two Geant4 models and REAM showed a good agreement on all the parameters we tested. GRRR also was also found to be consistent with the other codes, except for the electron energy spectra. That is probably because GRRR does not include straggling for the radiative and ionisation energy losses, hence implementing these two processes is of primary importance to produce accurate RREA spectra. Including precise modelling of the interactions of particles below 10 keV (e.g. taking into account molecular binding energy of secondary electrons for impact ionisation) also provided small differences in the recorded spectra.