I can't see your screen, nor do I know at all what you're doing. I have no idea where you're looking, where you're clicking, etc. If you need help you'll need to spend some time providing a thorough description of your problem.
"My library" is entirely separate from any group library.
Is "My name in the group" a subdirectory of the group library in question ?
To transfer information from "My library" to anywhere in the group library, you need to copy it. Last time I checked, directory structures could unfortunately not be copied, so you would need to manually recreate subdirectories and then copy the contents of each subdirectory by copy/paste with your mouse.
Upgrade Storage
http://www.zotero.org/support/sync
but otherwise I'm not sure what your question is.
Is "My name in the group" a subdirectory of the group library in question ?
To transfer information from "My library" to anywhere in the group library, you need to copy it. Last time I checked, directory structures could unfortunately not be copied, so you would need to manually recreate subdirectories and then copy the contents of each subdirectory by copy/paste with your mouse.