Is is possible to backup MMS/Group messages?
No, only SMS. Theoretically it is possible, but can not say it will be working for 100%.
Not being able to back up MMS is a bit of a deal breaker. Those SMS backup apps that don’t do MMS were acceptible in 2010, but in 2015, the better backup tools all back up SMS/MMS at the same time. I like Avast, but given that it’s a “premium” app and not some backyard hack, I’m very surprised that it cannot back up MMS messages.
-Naz.