I suggest you reread the documentation on groups and groupings. Groupings is basically a way to create a new set of groups and then assign it to an assignment.
So, for your example above, you would create the new set of groups Group A, B and C but then assign them to a single grouping - which you could call Assignment 1 grouping for example. Then when you assign that grouping to the assignment in question, the groups are automatically set up. From here, you can have them submit as group.
You ALWAYS want to make new groups - changing groups that were used for previous assignments will wreak havoc especially if they submitted as a group. You can assign students to as many groups as you want. Then use the groupings to decide which set of groups you want for each specific assignment.