I'm currently having a course about representation theory in the context of quantum mechanics and during the course, the professor said something about projective representations that I wasn't able to find elsewhere. He said that while using normal (i.e. non projective) representation, the corresponding transformation can be seen as a 1-1 mapping within the group. However, when projective presentations are used, the mapping becomes $1\mapsto e^{i\phi}$ and thus explaining the idea behind centrally extended groups.
Could someone send me a link where this kind of information is explained? I'm sorry for the lack of context but I don't have the notes and I couldn't find this information elsewhere.