You can set up an automation rule under Admin > Workflows > Automations > Ticket updates > New Rule to send out satisfaction surveys to customers when the ticket is assigned to a specific group.
When an action performed by..
Agent
Involves any of these events
Status is changed > Any status > Resolved
On ticket with these properties
In tickets > If Group > Is any of > Support
Actions:
Send email to requester > (you can add the content of the notification email and insert the placeholder for Satisfaction Survey)