can you set up service manager - so that when an incident is assigned to an agent in a different support group - than the incident is assigned - it will update the incident with the support group that the new assigned to user is a member of?

Short answer: yes.

You can use Orchestrator or PowerShell to change the support group. Normally when an incident is reassigned there is a log comment in the action log. You must monitor for new instances of the class Trouble Ticket Action Log. Look at how the description is constructed on (re-)assignments.