To take out a CBG, you have to ask two questions, (1) What is the primary offensive weapon of a CBG? and (2) What is the foremost thing that the commander of a CBG is afraid of?
First one is easy, it is the MIG-29. If you have better fighter jets than MIG-29, half of your worry is gone. As for the second question, I would guess submarine attack worries the CBG commander most.
It happened during Falkland War where British had to move HMS Hermes and HMS Invincible further out away from South American mainland after receiving reports of Argentine SSK activities in the area.
Sinking an aircraft carrier is not easy, especially if it has powerful escorts. However, a mission kill would cause the CBG to retreat, thus eliminate the threat from the ocean side.
A mission kill is achieved when (1) the CBG lose its effective air power (ex. losing 75% to 100% of its fighter jets) and (2) the aircraft carrier is not able to launch and recover its airpower due to damages sustained to its runway, rudder, air-control sensors or the ship is listing due to underwater damage etc.