We use density functional theory to calculate the electronic band structures, cohesive energies, phonon dispersions, and optical absorption spectra of two-dimensional In2X2 crystals, where X is S, Se, or Te. We identify two crystalline phases (α and β) of monolayers of hexagonal In2X2, and show that they are characterized by different sets of Raman-active phonon modes. We find that these materials are indirect-band-gap semiconductors with a sombrero-shaped dispersion of holes near the valence-band edge. The latter feature results in a Lifshitz transition (a change in the Fermi-surface topology of hole-doped In2X2) at hole concentrations nS = 6.86×10 13 cm −2 , nSe = 6.20×10 13 cm −2 , and nTe = 2.86 × 10 13 cm −2 for X=S, Se, and Te, respectively, for α-In2X2 and nS = 8.32 × 10 13 cm −2 , nSe = 6.00 × 10 13 cm −2 , and nTe = 8.14 × 10 13 cm −2 for β-In2X2.