See also the official documentation:
Help Center > Group Owners Manual > Renaming groups

Renaming a Group

CAUTION:  RENAMING A GROUP HAS CONSEQUENCES. READ EVERYTHING CAREFULLY BEFORE PROCEEDING.

It is possible to rename a group in place. You will not have to re-subscribe your group members, and all subscription information and file/photo/message ownership is retained.

Renaming your groups.io group is straightforward:

1. Go to Admin>Preferences>Settings and change the Group Title and Email Address, as desired.
2. Consider updating the Message Footer and Subject Tag entries (under Message Formatting) to reflect the new name.
3. Click on the Update Group button at the bottom of the page.

The renamed group may not appear to "work" for awhile. This is normal. Due to the subdomain addressing used by groups.io, any change to the root Email Address has to propagate through the DNS entries (so the new group URL resolves into an IP address). This can take several hours. 

Please note that renaming your main group will also change the URL of associated subgroups.

Some Impacts of Renaming a Group

As noted above, changing the Email Address of the group gives it a new URL on the groups.io site. This has several consequences.

• Subscribers will have to update the Address Book of their email client to send to the new address (of course).
• Subscribers will have to update their browser Bookmarks so they "point to" the new group URL.
• Groups.io will NOT parse old group content for group-related links and update them to the new URL. Therefore, many of the links to group content in the Message Archive (and of course, any previously-sent emails), your home page, and your wiki will likely now point to never-never land (404 not found).
• This includes the reply-to address and many of the links in group message footers!
• If you have synchronised the group calendar to an email client (such as Outlook) or external calendaring system (such as Google) you will need to update that also. That may mean deleting and recreating the calendar details in the mail client.
• Several member notices contain the group name in the subject and message text when they are generated. You will need to manually update these.

These are all good reasons to avoid changing the group Email Address once the group has more than a few members/messages.