"You" can't actually "start" one. They spawn automatically from the ports. One CV/ task group per port. If it (the CV) gets killed, the task group will die a few minutes later (15?) and a new task group will spawn from the port.
If the port is captured, the new CV will belong to the capturing team. A good strategy is to locate the enemy CV, then capture the port. Then kill the CV, and the new one will belong to your team. If you kill the CV first, it will re-spawn near the port you are trying to capture. That could make the port capture more difficult. (As it will still be an enemy CV)
In order to "direct" a task group, click on "Task groups" on the clipboard. Then click on "Take Command". You may need to select one task group first, if several are listed.
Note- ANYONE (on your team) with a lower overall rank than you can take command of the task group away from you. This is one of the incentives to having a low overall rank.
MtnMan