Parent Groups and Sub Groups v Categories and Global Groups....

I want to implement the following scheme in my site. Ive tried but it doesnt work so I think my understanding is not correct.

I have created Group for all countries in the world.

I want each country to have a standard set of sub groups or local groups e.g Singapore - Physical Security

Only people who are a member of the Singapore Group can join the SG sub groups.

I then want a set Global Group that any one can join. These will be the same name as the Country Sub Groups eg Global - Physical Security

Member example

John is a member of:

Singapore
Singapore - Physical Security
Global - Physical Security

He cannot join:

Malaysia - Physical Security
Australia - Physical Security

Because he is not in the Malaysia Group

I think I dont understand categories and groups well.

How is this achievable?

Thanks a lot.