No, issues can only be assigned to users. A work around is to create a email distribution list in your email system, add an ID to JIRA that can't logon and assign it that email. Then everyone in the distribution list will get an email about the issue. HOWEVER, issues that aren't assigned to a real person and thus appear in the Assigned to Me filter tend not be worked because no one sees them in their queue. One alternative is to assign them to the group lead and they can reassign as they see fit to team members. AGAIN, issues not assigned to real people are ignored.
Another option is to have a group-picker or select list that says "this group is responsible". But however you cut it, "assignee" should *always* mean "the one single person currently responsible". Whenever you move away from that principle it *always* goes wrong, with group members assuming that another member of the group is responsible. It doesn't work.
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