Samuel,
I don't know if that would work, but it's worth a try:
1. Create a "forwarder" email address.
2. Create an FB account with that email, address, set it to only email notifications and group notification ON, plus set it to private account, no friends, and all that stuff.
3. Login to FB with that account, join the FB group, and set the group notifications to ALL.
4. create a .io login for that email address and add it to your group.
5. Setup a rule in that account's mailbox to forward just the FB group notification emails to the .io group.
If it works at all, it may do what you want, maybe with some additional tweaking on FB & group notification settings, and on the forwarding email rule, to ensure only group-related and not FB-account related messages get forwarded. ?
I'd be curious if it does work!
Cheers,
Christos