If you look to reality for an analog, I think you'd struggle adding player-controlled carriers into the game.
You don't commit carriers into a fight; they are extremely stand off. There would be little to do between sending fighters out and recalling them. I'll take some time and really flesh out some ideas.
My full opinion on carriers:
Movement, shields, and on-board weapons
Carriers are going to be big, that's a given. They should move slow, which should be fine because carriers should not be in the fight. Their launched fighters engage, and, ideally, a carrier should not be fired upon.
Large ships should imply large reactors, giving carriers dreadnought-level shielding.
Lastly, carriers should have small, close-range weaponry. Something comparable to the artillery's secondary weapon: light cannons to repel quick-attack strikers.
Player control versus AI
This is where I have a lot of opinions. With little to do between launching fighter waves and being stationed well outside the fight, player control could be rather boring. Having AI control over this could be effective, though mechanics would have to be implemented to allow other players to influence who and how the carrier's fighters attack. Simple coding could be implemented to handle point defense of the ship proper.
The other alternative, I think, may require more game development. There could be implemented a sort of Fleet Commander position, who operates "within" the carrier, controlling its functions but also being able to interact more indirectly with the other players. For examples, this commander may be able to interact with mini-map and HUD beacons/tags, vector fighters to targets via the map, or even go so far as control environment elements (maybe static turrets, moving structures, etc), but that may really start compounding the complexity of adding in carriers.
The other alternative is to have the carrier controller interact more with the fighters themselves. Instead of just deploying the fighters, the player could pilot them, switching between egressing fighters, spent from battle, to ingressing bombers. This would cut down on the "boredom" a carrier pilot may experience.
I envision the actual launched craft mechanics operating similar to Battlestations: Midway. The controller would have access to multiple types of crafts to be committed through a set number of hangers. Types could be numerous: fighters to protect other craft, bombers to target ships, repair drones to buttress your teammates defenses, or perhaps a more exotic options. Each type would differ in their speed, strength, shields, and formation numbers. Each hanger could be selected to launch any type, allowing the carrier to keep a level of flexibility to remain relevant at any stage of the battle. However, there would be a set number of hangers to limit what is committed at one time, as well as a hard limit to the number of crafts the carrier has over the course of the battle. Commit nothing but bombers with no cover early on? Rough, you've run out and only have fighters left. As sorties return alive, their numbers would be returned to your totals.
Finally, these craft would likely warp into and out of battle, given the distance from the fight the carrier should be.
Would love feedback on these ideas.