didn't get your tag for some reason.
There are only one way to sink an aircraft carrier. That is to break into the complex defence system that cover sea, air and information.
An aircraft carrier is protected in 4 tiers defences. The first tier is information.
Information is about threat detection and threat management, it may not be preform by the Carrier Group itself, it can be done by any of the ISTAR platform, ground radar, AWACS, maritime surveillance aircraft, naval surveillance ship, escorts and so on. The first battle is for the attacker to by-pass the detection, and the defender to sniff out attackers, both surface and sub-surface.
The second tier is the defence from the on broad air groups, they can reach out over the reach of the escort ship defences, and they can also detect and attack any threat present in the immediate area, each carrier carry about 50 F-18E/F and in between all those aircraft, they carry 200-400 AAM/ASM as a whole. Couple with AWACS like E-2D and other AWACS/AEWS platform, their job is to seek out the enemy before the enemy can enter the next phase.
The third tier is the defence form escort ship, a CBG would usually travel with 5-6 DDG and 1-2 CG, which mean at any time, the whole 220nm radius will be covered with 700 - 900 cell Mk41 VLS, which can be potential launching a mix of AAM like SM-2 and SM-6, ASM like Harpoon and BMD like SM-3 Travelling with a CBG would also include some Submarine and other assorted ship. Also with 5 inch gun, 3 inch gun and any sort of CWIS, Bushmaster and Machinegun.
When the threat can penetrate all that, then it will the forth tier, which is on broad ship defence. Which will be a mix of sea sparrow, RAM launcher, and CIWS.
That is what it take to sink a Carrier.