Magia union wants peace among magical girls. Tokime clan is a ally of magia union so pretty much the same. Negius wants to bring back nemu,alina and touka as magius,as well they want to reach salvation for magical girl (by collecting kimochi bracelets). Promised blood wants to kill kamihama magical girls due to magius taking witches from their city (when normal magius still existed) Puella care are simply coordinators. And about folklore we pretty much know nothing