Create forwarding email address for mailing to members of a group
This problem has two parts - getting the email into Civi, and sending it out again to people in a group.  The first part is definitely going to be done with Inbound Email Handling if the person will be using their email client.

For the second part, I like Tim's idea of integrating with Sympa or Mailman, but unless you need the full-blown list management software, it seems like overkill.

Pete's CiviRules answer should be correct; however, just this week I tried doing almost the exact same thing (redirect incoming SMS to a team's email) but I couldn't get the tokens to work.  My client was on a tight budget so rather than troubleshoot I just repurposed an existing extension.  I would try this CiviRules approach first.

Another approach would be Scheduled Reminders (Administer menu » Communications » Scheduled Reminders).  You could set a reminder to trigger on an "Inbound Email" activity, and use the Select Group dropdown to pick the group.  Your token would just be {activity.details} to pick up the message body.  If you do this, you should make sure your cron runs reasonably frequently.  You would also want to increase the frequency of the Process Inbound Email and Send Scheduled Reminders jobs under Administer menu » System Settings » Scheduled Jobs.