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 (alpha and beta) 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 n(S) = 6.86 x 10(13) cm(-2), n(Se) = 6.20 x 10(13) cm(-2), and n(Te) = 2.86 x 10(13) cm(-2) for X= S, Se, and Te, respectively, for alpha-In2X2 and n(S) = 8.32 x 10(13) cm(-2), n(Se) = 6.00 x 10(13) cm(-2), and n(Te) = 8.14 x 10(13) cm(-2) for beta-In2X2.