Another possible explanation for this is that they simply signed it in secret without the government's knowledge, but this is made unlikely by how ingrained the Illuminati is in Area 51, which is a top secret government-run facility. Many groups believe that the Illuminati are evil, and actively fight against them. The members of the Illuminati themselves believe that they are simply doing what is best for humanity. Thus, whether or not they are benevolent is ultimately subjective.
The secrecy of the organization leads to attempts to fight against them usually being fruitless, and for the most part, pointless.
The Illuminati's forces are first encountered directly after Ethan Cole gains the ability to shift into a mutant. They are found for the first time right after a room full of Leapers. The Illuminati's leadership, barring Mr. White, are never encountered in game. Instead, their main fighting forces, the Black Ops , generally do the work for them and end up being the foes the player fights throughout Area 51.
The Illuminati's main fighting force works in what appear to be highly organized squads. They use weapons that can be often seen in Area 51, including the M-170 Shotgun, Viper assault rifle, BBG-11, and Sniper Rifle. The Illuminati forces are never seen using a Scorpion Pistol or MC-11 Meson Cannon. It could be assumed that this disparity in usage is either because the Viper and BBG-11 are more common/standard issue, or that they are simply preferred weapons for the Illuminati.