Google Groups don't seem to have a setting for notifying (and sending a copy) whenever a message is sent from the group. In other words, only the person that sent the message from the group knows it was sent, and the other group members don't know it unless they manually enter the group and check for messages.
Is it possible to do it through Google Apps Script? i.e. trigger an event every X minutes to catch only outgoing messages from the group, and then send them to the group?
I don't think Apps script is the best option because as far as I can see when emails are sent from the groups interface they do not show on the "Send" items of the sender, there might be a way to do this through Apps script however I could not find a way to account for those email in documentation.
An alternative (if you are using Google Workspace) would be to create a routing rule (There is another alternative but I will focus on this one at this moment)
from the admin console, you can do so by going to Apps> Google Workspace> Gmail> Routing (The last option), then click "Configure" under "Routing"
Then you just need to add the following settings:
Enter a short description at the top
Select "Outbound" and "Internal - Sending" under "Email messages to affect"
Scroll down almost all the way until you find "Also deliver to" and check the box
Click "Add"select "Basic" and type in the groups email, that will be the recipient of the emails but we need to make sure to delimit the sender so it only affects the messages sent by that group so we need to:
Scroll all the way down to a little section called "Show options" it looks like this:
set "Account types to affect" as "Groups"
Set "Envelope filter" to "Only affect specific envelope senders" leave the dropdown with "Single email address" and under "Email address" type in the email of the group
Click "Save"
The other alternative would be to use this setting from the groups interface:
however I am not sure how the setting works or how it would affect your team.
In any case I hope you can find this useful!