The send e-mail function within the Action Rules area does not provide the proper tags to send the information you would want to the external e-mail address. Unless you can fit all of that information in the Subject text box on the ticket.
You can use an Action Rule to modify a ticket and add a CC or BCC e-mail address to the ticket. You would have to select the correct criteria to make sure the right ticket has the address added at the right status (i.e. When status equals Resolved, or Pending, etc.).
Like Rick says, the answer is "Yes you can"
In Setup, select Processes > Action Rules
Create an Action Rule Info, where the Rule Triggering = "only when updated by a Tech"
When criteria evaluation results in a match, apply action… = "only if criteria did not match before the Ticket was updated"
Select the Criteria TAB set
Tickets matching ALL of these conditions = Request Type is <pick your request type from the drop down>
… and ANY of these conditions:
Status is "Closed"
any other conditions you want to select
Then select the Actions TAB
Select Send E-mail
complete the Recipients, Subject, Template information the way you want to email to appear.
SAVE, and you should be good to go.
I may have just answered your question here: Custom ticket variables/fields in emails and/or ticket notes? The difference is that you could include a custom field or text box content (populated by your team) to forward to the different support group in the email after the ticket is closed. Be sure to add a note in the email to reply to the outside email address as a reminder so that they don't just hit reply.
Hope this helps.