The shape and evolution of the energy spectra of the thunderstorm ground enhancement (TGE) electrons and gamma rays shed light on the origin of TGEs, on the relationship of modification of the energy spectra (MOS) and relativistic runaway electron avalanche processes, on the nature of the seed particles, and on the strength and elongation of an atmospheric electric field. However, till now the measurements of energy spectra of TGE electrons and gamma rays have been rather scarce. For the first time, we present differential energy spectra of gamma rays in the wide energy range 4-100 MeV for five TGE events detected in 2012-2013 at Aragats. We use the special technique of electron/gamma ray fraction determination to select TGE events with very small contamination of electrons. The network of large Nal spectrometers located 3200 m above sea level measured energy spectra of gamma rays. The power law indices of "small" TGEs are rather close to the background cosmic gamma ray spectrum ( gamma ~ -2); thus, we may deduce that these small events are due to MOS of cosmic ray electrons in the electric field of a thundercloud. Larger TGEs measured by the Nal network and the two largest TGE events earlier recovered from energy releases in a 60-cm-thick scintillator have much steeper energy spectra typical for the avalanche process in atmosphere. The classification of TGEs according to intensity and gamma ray spectral index pointed toward two main mechanisms of the TGE gamma ray origin: the runaway process and modification of electron energy spectra in the thunderstorm atmospheres.