https://developers.facebook.com/docs/graph-api/reference/user/groups for Graph API Version v3.1 appears to indicate a solution is possible if the App has gone through App review or if the App is in Development.
https://developers.facebook.com/docs/graph-api/reference/v3.1/group indicates another possibilty with the permissions:
groups_access_member_info — Enables your app to receive member-related data on group content.
publish_to_group — Enables your app to post content into a group on behalf of a user.
user_managed_groups — Enables your app to read the Groups of which a person is an admin.
With that said however, The user_managed_groups permission is not in the list of permissions which can be requested and even in Development mode with a Test App I am getting an error message:
English Quiz - Test1
Ethan 100029439245680
==== Query
curl -i -X GET \
"https://graph.facebook.com/v3.1/100029439245680/groups?access_token=<access token sanitized>"
==== Access Token Info
{
"perms": [
"user_friends",
"groups_access_member_info",
"public_profile",
"basic_info"
],
"user_id": 322852501618049,
"app_id": 867918616743637
}
==== Parameters
- Query Parameters
{}
- POST Parameters
{}
==== Response
{
"error": {
"message": "Unsupported get request. Object with ID '100029439245680' does not exist, cannot be loaded due to missing permissions, or does not support this operation. Please read the Graph API documentation at https://developers.facebook.com/docs/graph-api",
"type": "GraphMethodException",
"code": 100,
"error_subcode": 33,
"fbtrace_id": "EvrL0yxzs/q"
},
"__debug__": {}
}