Early on in the series we learn that the aliens are answerable to a supreme leader. He is the one referred to as The Big Giant Head and is admired and feared in equal measure by his alien drones; although one of them is heard to remark that he only got the job by kissing the Big Giant Butt.
The Big Giant Head's powers are enormous. It is frequently inferred that his prerogative allows him to cancel the mission at will and summon his alien drones back to the home planet. It is even suggested at times that he has power of life and death over his minions. We are also led to believe that he is not just a mighty leader on his own planet but that he commands vast galaxies. He is in possibly something akin to Master of the Universe (or a vast part of it at any rate).
In season four we finally get to meet The Big Giant Head, when he arrives on Earth in the form of William Shatner for a tour of inspection. This proves to be the first in a series of visitations from above.
Each visit brings new problems in its wake and prove hectic where Dick, Sally, Tommy and Harry are concerned. For one thing, the Big Giant Head proves adept at breaking all his own rules and protocols where alien-human interaction is concerned. He even manages to get a female member of the human species knocked up.
Adding to their woes is the onus placed on the four aliens to appease an overbearing, tyrannical, indeed meglomaniacal master; at the same time protecting the inhabitants of Earth with whom they have bonded; all the while maintaining the deception about their extra-terrestrial origins.
The final appearance of The Big Giant Head suggests a somewhat mellowed (though still overbearing), possibly hen-pecked character. It is at this point that he reveals to Dick that he is in fact his father. This prompts ;Harry to point out that the High Commander is a Dick Head - and who said he wasn't smart.