How to export Group module entities with the Features module?
Have a look at the UUID Features Integration module, which provides a mechanism for exporting content (nodes, taxonomy, fields) into a features module. It depends on the UUID and Features module. Some more info aboout this module(from its Community documentation):


  Drupal's core entities are supported by default within the UUID module, but if you want to export content contained within custom entities, you will have to implement UUID support using their API.


The challenge with the UUID Features Integration module is that its documentation is not complete (yet). So for adding UUID Features support for custom content entities (such as Groups), refer to the module's includes/ folder and use one of the existing implementations as an example.

2 more documentation issues that may be of some help also:


Issue # 2310115.
Issue # 2236497.