If you need to add an email address for someone who does not have an @uah.edu account to a Google Group that you manage, it is necessary for the Group settings to reflect that such membership is allowed. To verify or change this, go to Group Settings > Permissions > Basic permission > Allow members external to this organization > Allow new users not in uah.edu.
If you check the All new users not in uah.edu option among Basic permissions, anyone who has manager access for the group will be able to add members outside of UAH.
If this option is cleared, then people whose email account is on another domain (such as @gmail.com) will be unable to join the group.
Click here to read more about Google Groups. Section 5 covers the details of managing a Google Group.