Task list has email enabled on task assignment, tasks can be assigned to multiple people or groups. When I create a task and assign it to myself, I get mail. When I update the task to assign it to a group, I get mail but not the group. When I create a task and assign it to a group, I don't get mail.
The site and list are open to everyone in the organization. I've toggled the send email setting on the list. IT modified the Active Directory groups being used for notification to disable the "Require that all senders are authenticated" setting.
I have full control on the sites for my organization, but I do not have SharePoint admin access. The company's SharePoint admin moved on a few weeks ago and we don't yet have a replacement.
ETA: I checked with IT and the lists are already security groups.