Group email question

There are couple of options you can use:

1. Have all your support team members enable conversation view. Ref: https://support.office.com/en-us/article/view-email-messages-by-conversation-0eeec76c-f59b-4834-98e6-05cfdfa9fb07

By doing so, when a team member replies to a customer (assuming the group email address is not removed from loop) other team members will be able to see the chain thread in Outlook/OWA.

2. Create a Shared mailbox for the group email address- shared mailbox do not require license. All the team members can be granted access over this Shared mailbox, and this will show up as an additional mailbox on their Outlook client. You can create sub-folders under this Shared mailbox for team members to move the mails to different sub-folder when they've taken ownership (sub-folders can be created for each team member of need be).

That way, if a customer mail is in Inbox it has not been actioned yet; if it has been actioned the mail would be in a sub-folder and not in Inbox anymore.