Adding a custom variable to the group object

Hi Guys,

I'm back into deep customization and I have never really worked on this part before. I want to add a variable to the Group object.

A)Thus I would need to first get a cue at HOW I can first do this programmatically : I mean do I just have to add a column in the database then do a $category->get('NEW column name') after adding the variable to the TABLE group object ?
i) add column table in database
ii) add variable to the table group object
iii) get/set the variable through the general $category object

B) I would like to get which file is relate to the screen enclosed.
and how the get/set method is set into that file so I can add this custom var to the UI.

Thanks Guys, this would be a great piece to add to your dev book btw