If it's a Joomla module then that's easily accomplished using Joomla's built in loadmodule behavior in a Custom HTML field with content plugins enabled (see parameter under parameters tab). If its rendering is done entirely through a component view you'd need to explore using something like Components Anywhere to try and render it using a shortcode (similar to loadmodule) in a Custom HTML module. If they have their own shortcode then that would also work same as loadmodule does.

Ah, sorry. Misunderstood what you were asking. The only way to add a new tab to GJ is via API, but you can shortcut that by using CB Auto Actions. The below will render a new tab in CB GroupJive.

Global
Triggers: gj_onBeforeDisplayGroup
Type: Code
User: Self
Access: Everybody
Conditions
Field: Custom > Value
Custom Value: [var2_published]
Operator: Equal To
Value: 1
Action
Method: HTML
Code:Output
Display: return
Layout:Method: PHP

As for the Code under Action it can basically be whatever you need. If you need PHP to call a 3rd party extensions API to render that extension then you'd do that there. If you wanted to substitute in a Joomla module then turn on "Content Plugins" under Parameters and use a loadmodule shortcode for example.

Replace MY_TAB_ID with a unique ID for your tab and replace MY_TAB_TITLE with whatever you want the tabs title to be.

If you need this unique for each group you'll need either 1 auto action per group or in your Code check for group when outputting whatever you need to output. You can get the group id using [var2_id] to either condition against as follows.

Field: Custom > Value
Custom Value: [var2_published]
Operator: Equal To
Value: GROUP_ID_HERE

Or within your code as follows.

Method: PHP
Code: