In Ga2-based Heusler alloys Ga2XCr (X = Mn, Fe, Co, Ni, Cu) the tetragonal distortion, electronic structure, magnetism and phonon dispersion have been studied by first-principles calculations based on the density functional theory. The volume-conserving tetragonal distortions of the cubic Ga2XCr show that Cr atom makes the greatest contribution to the total magnetic moment. No martensitic transformation has been found in Ga2FeCr, Ga2CoCr and Ga2CuCr. For both Ga2MnCr and Ga2NiCr, the tetragonal phase is lower in energy as compared with the cubic phase. Ga2MnCr and Ga2NiCr have the lowest total energy at c/a = 1.28 and 1.11, respectively. Correspondingly, the energy difference ΔE between the cubic and the tetragonal phase is -8.26 meV in Ga2MnCr and -6.14 meV in Ga2NiCr. For Ga2MnCr and Ga2NiCr, calculations of electronic structure and phonon dispersion reveal that a sharp peak near the Fermi level will lead to a structural instability by increasing the energy of the system, which can result in a broadening in the energy range due to hybridizations between 3d electrons as well as the potential structural transformation. With proper c/a and ΔE a potential tetragonal martensitic transformation can be expected in Ga2MnCr, the phonon dispersion of which further shows that the acoustic modes tend to be softened.