Not what you’re looking for, but:
The “Automatic” field is the group’s automatic_membership_email_domains property. As you can see, it’s not in the attributes list.
The “Automatic” field that you’re seeing in the payload indicates whether or not the group is an “automatic” group. Essentially, the built in trust level and staff groups are automatic. Custom groups that you create are not automatic.
It seems like it would be easy to add the automatic_membership_email_domains field to the group webhook serializer. Also, this information should be available without having to look at the source code.