Pyramid Head and The Entity
Ok so I know this is a bit out there, and this is the first time I've mad a theory like this so please don't shoot me down too hard. But from what I've gathered when it comes to the Entity, as powerful as he is, he isn't omnipotent, he only controls those within his pocket dimension, however he is meant to have total control.
Now hear me out, I believe that while the entity was able to entice Pyramid Head into the fog, I don't believe he has total control over Pyramid Head as he does other killers. Most he has to bargain with but he can't bargain with Pyramid Head, and the way Pyramid Head operates seems vastly different from other killers. Note his rites of judgment appearance and how it is described. "The Great Knife splits the ground, giving rise to an infernal construct and spreading agony to all who approach." That gives the impression that it is more than the entity.
Where Pyramid Head treads it seems he can bring the other world into the entity, the other world is an entity all it's own, now one could say that the entity allows this to happen, but if that were the case why not it simply be how he allows Freddy to pull survivors into the dreaming world?
My theory is that the Entity is more than aware of the otherworld and as such tries to keep himself separate, but that is why when Pyramid Head leaves the trail it looks like a wound in the ground and when Pyramid Head or survivors touch it the entity loses some influence and control over them.
If we consider the cage of atonement something all of Pyramid Head's and by extension, the other world's design, then that would make sense how Pyramid Head can create that when the survivors run through his trail. Again if we take it as the trail is the influence of the other world within the entity itself. The entity isn't omnipotent, and the power of the other world allows Pyramid Head as well as survivors to in a way, break connection with the entity. Even one of the items of Pyramid Head, the obsidian goblet, allows him to COMPLETELY break his connection with the entity by becoming undetectable as long as he remains on his trail.
This would also allow him to execute survivors when he sees fit as he doesn't obey the entity and when the other world has it's clutches into a survivor (through torment) it allows Pyramid Head to disregard the regular rules of sacrifice with his own methods including executing the survivors on his own terms without need of the entity telling how to kill them.
Of course all this can be chalked up to game mechanics, but I like looking at things from a different perspective, and really it isn't all that far fetched when we realize that not only is the Entity not omnipotent, but other killers have been shown to be capable of wresting control from the entity before such as through the Huntress's perk Beasts of Prey where the flavor text reads "your lust for a kill is so intense that your connection with The Entity is momentarily lost, making you totally unpredictable."
This means that it isn't impossible for humans (and others) on their owns to break control, now imagine a being that is from an alternate plane of existence that thrives on human torment and is born of their very guilt where the realm itself can shape the world however it pleases in the eyes of every human who witnesses it to the point where two people can witness the other world completely differently from one another.
My "in conclusion" is this. The entity is able to hold Pyramid Head, but out of all the killers has the least control of him as Pyramid Head's connection with the other world is much stronger than it is with the entity, and the entity only has control of Pyramid Head while their goals are the same, but if Pyramid Head wishes to act differently, then the Entity has little control to stop him.
(again sorry for the very long post, I had just been thinking about this for a while and I just really wanted to say this to see how you all would react to it. Please don't throw hate, but I will accept genuine criticism because I doubt this theory is perfect in any way.)