Groups in Blackboard are used for organizing students. There is no group submission process.
If you are using Groups or Group Sets in Blackboard, you can use the dropdown, which appears at the top of the Assignment Inbox, to organize your students.
Groups and Group Sets are only available for customers on LTI 1.3. Speak to your administrator about upgrading from LTI 1.1 to LTI 1.3.