Is there a way to add objects to an empty group? I think it is a special case. If I remove objects form a group that group still exists without objects. So I can add objects to that group via Properties / Object / Groups / Add to Group button. The list will show the name of the existing groups so I can add object to one by one.

So the question: how can I add more than one objects to an empty group (not one by one)?
Select your objects you want to add to the empty group and run Python script:

import bpy

group_name = "James Bond"
for o in bpy.context.selected_objects:
    bpy.data.groups[group_name].objects.link(o)