You can route emails sent to a specific support email address to a group directly from Admin--> Email.
However, if you have the tickets being shuffled between groups on a regular basis, you can make use of the Dispatch'r to add a tag using which you can filter the tickets.
Navigate to Admin--> Dispatch'r--> Create new rule.
Condition:
"To Email" "is" "The support email address"
Action:
"Add Tag"