We have recently refreshed our branding across our offerings and changed the names of our pricing plans. If you have signed up before Aug 9, 2021, please click Previous plans to view your applicable plans.
We assure you that this change will not impact your product experience, and no action is required on your part.

Yes, this would be possible if the latter group has automatic assignment enabled for it. 

Please navigate to Admin > Team > Groups > click on edit next to the group you are moving the tickets to and Enable "Automatic Ticket Assignment."