Foundation XYZ is an evil organization delicate to the goal of ruining the Symphogear Universe and turn it into a second Ultimate Tokusatsu Chronicle Universe ... WITHOUT the hero.
Foundation XYZ can make various type of monster that once appear in the Ultimate Tokusatsu Chronicle Universe and intergrate them into the world of Symphogear. They're mysterious in their method and the nature of their leader is unknown. As some of their leader even appear to be on the side of good before they reveal their true nature.
In order to achieve their goal, they often kidnap people from poor, undeveloped country and turn them to "Modified Human". Sometime they use Dimension Modifying machine to fuse many aspect of other parallel world into the world of Symphogear but that do admittedly take alot energy and resources to do.
In short, they will use any method available to them for the purpose of sabotaging the world of Symphogear.