A series of GW calculations for zinc-blende Zn, Cd, and Hg chalcogenides (S, Se, and Te) is presented. The resulting quasiparticle gaps are 0.3-0.6 eV smaller than in experiment. The cation semicore states remain similarly underbined as in previous GW calculations for IIB-VI materials. It is shown that application of the plasmon-pole model for screening leads to systematic and qualitative errors: the band gaps result larger and the occupied band widths undergo expansion, instead of contraction. Several steps, like including off-diagonal matrix elements of the self-energy, updating eigenvalues in the Green's function and screening, and using the special vertex corrections, are examined. We also propose a scheme to treat core corrections in the case when Zn2+, Cd2+, or Hg2+ pseudopotentials are used in a GW calculation.