I take it by "Group" you mean a Mailing List?

You cannot "transfer" a contact from an Address Book to a Mailing List. A Mailing List is a subset of its parent Address Book. Anything you add directly to a Mailing List will create a matching parent entry in the parent Address Book.

Add new contacts to the address book. If you want to set up a Mailing List, add contacts to it from the address book.

Think of a Mailing List as adding tags to contacts in your address book. So you might have a "group" for your family. Another "group" for members of a club. Some family members may also belong to that club, so those contacts would have two tags. But then you have only one address book contact entry to maintain, rather than having to look for separate entries in multiple Mailing Lists.

If you really want separate storage for certain groups of contacts, then look at File|New|Address Book. But be aware that Mailing Lists are designed to help you efficiently address messages to the same bunch of people, time and time again, so it's good for clubs and newsletters and so on. Address books don't help so much with doing this, because they don't allow you to represent all those people as one address entry, as does a Mailing List.

<tl;dr>? Mailing Lists are NOT address books.

Many thanks for the explanation altho I haven't had time yet to assimilate it fully, knowing how Thunderbird works will sure help! Tom

Many thanks Zenos, the problem I had was between two 'e' mail systems that use slightly different language, so in one "group" means something else in the other! But safe to say your answer was exactly what I needed, many, many thanks.