You can only assign issues to users. The work around is to create a user with a distribution list as the email. Don't put them in the jira-users group so they don't use a license. You can then assign it to that user and everyone on the distribution list will receive an email. WARNING: Issues not assigned to a real user tend to be overlooked because the don't appear in the 'assigned to me' built in filter.
Can you elaborate more on what do you mean by go to a specific group? Do you mean:
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