When it comes to notifications, People take priority over Groups. This means if a shipping label contains a Person's name and a Group's name, then the Person will be notified instead of the Group Contact.
Group Contacts only receive notifications if items are not addressed to specific People in their Groups. If a Group Contact would like to receive notifications whenever a member of their Group receives an item, then you can update every Group Member's contact information to include the Group Contact's contact information.
(If you do this, we strongly recommend that you do not delete Group Members' personal contact information. This way, Group Members will still receive their own notifications and Group Contacts will be kept in the loop.)
Additionally, make sure everyone in the Group has the same Preferred Contact Method if you decide to do this. This is the only way multiple notifications about items for Groups will be sent successfully.