The scenario is this:
I have a Google group; let's call it firstname.lastname@example.org. Anyone can send email to this group. The group has 2 members, personA@company.com and personB@company.com. Both personA and personB has email@example.com set up as an additional email address in Gmail settings with "Treat as an alias" checked. This means that both personA and personB gets mail sent to the support group in their own personal inboxes, and can reply as "firstname.lastname@example.org". So far so good.
The issue I have is that if personX@whatevermail.com sends a mail to email@example.com, and personA replies to this mail using the firstname.lastname@example.org alias, personB gets no indication that personA has replied to the mail, personB can't see the mail that personA sent as a reply to personX anywhere. Vice-versa obviously also applies if personB replies.
How can I fix this so that the group members can see all sent mail from the group, regardless of which group member sent it?
Both group members also use the Outlook mail client if that makes any difference.