The Air Wing Commander's duty station is in his cockpit, flying his fighter plane, leading his Carrier Air Wing
Actually, I am not sure if the CAG actually flies anymore. I think his duty station is in the CATCC (Carrier Air Traffic Control Center), but I might be wrong about that.
Not wrong but that's not all there is to it. CAG still does fly- it's a flying billet. However, under the combined warfare commander concept that's been around since the Reagan era, the CAG is also the 'Strike Warfare Commander' for the battle group and has a whole staff assigned to him for strike planning purposes. This position is more about power projection (offensive operations) than force protection. So the modern day CAG wears a couple of hats, but he does still fly.
There's another guy aboard ship as well called the 'air element coordinator' or something like that whose job is to parcel out the entire battle group's air assets (to include the carrier air wing) on a daily basis to the various warfare commanders in the battle group based on mission requirements. For example, say on any given day you are going to send a SEAL team down range on some sort of mission, and there is a chance they might need close air support while they're at it. Simultaneously, you need to conduct a search for a possible threat submarine and maintain a CAP against a counter-attack along the threat axis in the direction of your adversary. At the same time, one ship needs critical supplies delivered, another has a planned medical transfer after a casualty, and it's mail call day.
I'm not sure who plans the SEAL mission, but let's say they need at least four helicopters- two for transporting SEALS and two supporting CAS gunships.
CAG (Strike) would be responsible for coordinating the fixed-wing CAS portion of the SEAL mission. This requires a package of F/A-18 Superhornets as strikers and tankers, and an EA-18G Growler for jamming / EW.
The USWC (Undersea Warfare Commander) is responsible for the ASW problem, and wants as many helicoptors as he can get fitted out with ASW loads to provide his ASW screen. This officer is probably the CO of one of the battle group cruisers or destroyers.
The AAWC (Anti-air warfare commander) is responsible for force protection against air attack. He is one of the Aegis-equipped cruiser skippers. He's got the missile defense systems on the battle group's ships, but wants 12 hours worth of F/A-18 Superhornet fighters (armed for CAP) on station, with tanker support on a ready alert status- say a minimum of 4 fighters on two stations at a time, with at least one plane on deck as a reserve tanker.
Someone (The ATO?) wants helicopters for the vert-reps, medevac, and mail runs. Today he's probably going to get the short end.
All these asset requirements have to be coordinated, prioritized, and parceled out. On any given day the carrier squadrons and small-ship aviation detachments have a certain number of airframes and crews available for use. Some planes are always down for maintenance, and crews work on duty and crew rest cycles which mean they aren't all immediately available to use 100% of the time- squadron scheduling officers and operations department types keep that sorted out at the squadron/ detachment level. The air element coordinator is the poor guy/gal who works for the admiral and gets yelled at when everyone doesn't get exactly what they want. They just make it happen the way it needs to happen to get the job done.
Anyway.
