Hi, I think it's a problem more SharePoint-related than a Nintex one..
my suggestion is that in user information list of the current site collection the group is stored with the old display name..
If you try to set that group in a standard sharepoint user group field and save the item, does it shows you the old or the new name?
If it shows you the old name, I'll try to follow this suggestion (even if it's for SharePoint 2010): Updating SharePoint 2010 User Information « SharePoint Automation
The powershell script is designed to update a user display name, not a group one, but I think the script could be adapted to your situation
Hope this could help
Giacomo
Most likely this is a cache on the Outlook side. Outlook will apply names as it knows them from it's frequent contacts or global contacts list. When I had a colleague change name once I opened a new mail in Outlook, typed their name in the To line, then click the X next to their name when it was prompted. This deleted it from my frequent contacts list. I then sent a new message using just the email address and when they replied it had the new name.
Not sure if this applies, but could be worth a try. I've seen this article on the subject Remove suggested name or email address from the Auto-Complete List in Outlook Web App - Outlook