Recently I read the chapter 2 of Weinberg's QFT vol1. I learned that in QM we need to study the projective representation of symmetry group instead of representation. It says that a Lie group can have nontrivial projective represention if the Lie group is not simple connected or the Lie algebra has nontrivial center. So for simple Lie group, the projective representation is the representation of universal covering group.

But it only discuss the Lie group, so what's about the projective representation of discrete group like finite group or infinite discrete group? I heard it's related to group cohomology, Schur's multiplier and group extension. So can anyone recommend some textbooks, monographs, reviews and papers that can cover anyone of following topics which I'm interested in:

How to construct all inequivalent irreducible projective representations of Lie group and Lie algebra? How to construct all inequivalent irreducible projective representations of discrete group? How are these related to central extension of group and Lie algebra ? How to construct all central extension of a group or Lie algebra? How is projective representation related to group cohomology? How to compute group cohomology? Is there some handbooks or list of group cohomology of common groups like $S_n$, point group, space group, braiding group, simple Lie group and so on?