Can I copy or move contacts from one group to the other?

Surely, you can. Here are the steps to copy/move selected contacts from one group to the other:

1. Open the group from which you wish to copy/move the contacts.
2. Search the contact using the search bar given at the top.
3. Checkmark/select the contact that you wish to copy/move.
4. Click on the "Copy To" or "Move To" button at the top bar to select the groups to which you wish to copy or move the selected contacts. 
5. Press Done if you are copying the contacts, moving contacts will only need you to select the group. 

Contacts have been copied/moved successfully. :)