You can add and remove the groups you wish to show at the bottom of the Group Memberships page in your UserCP. To do this, scroll down to "You may choose to be identified as a member of the following groups." and select the group you wish to be shown as. Your primary group (Pirate) will always show, along with some other tags including guild officer tags, arcade/trivia winner*, and clan leader.
You may only show one of the groups you can join though. These include any member guild tags, ranter, and glitch pirate.
*If the tag comes with a special name such as Trivia/Arcade Winner, Moderator Tags, or Clan Leader tags, showing one of the groups you can join WILL shut off the special name, however the tag will remain.
(Thanks to Jack I for that answer
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