Reviewing through your description, it's possible that a duplicate account has been created for this user once their email was changed. To confirm, I would suggest searching your admin console >> people page for both the old and new email addresses to see if multiple accounts are present. If so, having the user access via the older account might allow them to access the group.
That being said, if this does not resolve the issue, I would recommend submitting a case in your customer group ( MyJive Secret Support Group Administration FAQ ) as a next step for further investigation by a member of our support team. When submitting the case, please also include the following:
1. current email address for the user
2. previous email address for the user
3. link to the group the user is wanting access to