Frontend fields edit in userlist

I currently have the CB Core Ajax plugin installed, allowing frontend Superuser editing of fields displayed in a userlist or profile.  Works great.  

I was wondering if this editing permission could be assigned to a different access level or group (instead of only a Superuser)?  Could this be controlled at the field tab level or the individual field?

Where would this permission be managed?    

CB Moderators have permission to edit profiles as well if enabled in CB > Configuration > Moderation. You can then make other users CB Moderators via the Moderator View Access Level setting by creating a Joomla view access level to include whatever Joomla usergroups should have moderator permission. Next simply give that usergroup to the users you want to have moderator permissions. They should then be able to edit profiles among other moderator permissions.

Using your instruction, I got it going exactly as needed.  Thank you!