If the target has plenty of aircraft it might try to overwhelm the carrier group.

the first and formost priority of the Carrier wing is to protect itself and then conduct strike missions , there is always a section of the force ready to launch into CAP to protect the wing. Furthermore try to imagine in the current scenario what would a strike package need to be to get past with a dozen or so(more but i’m taking the lower range) F-18E/F’s and F-18C ,D’s (f-35’s in the future) , a huge search and track active and passive detection system , AEW assets , AEGIS , ESSM , Phalynx , dedeicated sat. coverage and reports from other assets and then factor in that it would only be logical for the USAF and USN to conduct dedicated Strike missions (using host of stealthy and non stealthy assets,cruise missiles etc ) to destroy the enemies capability to launch large offensive packages to threaten anything , and even then the airspace would require the clearence of the USAF’s and USN’s assets that is to stay up and do anything worthwile the enemy would have to destroy USAF and USN assets that are dedicated to just one thing – destroying anything that goes up in the air capable of doing any sort of damage .

Remember the USN never goes to war alone .