Ticket group is set to "Not Assigned" when picked up by a Technician

I've noticed when a ticket is picked up by a technician that is not associated with a group, it marks that user as the technician and sets the group to "Not Assigned". Generally our technicians have their view set to show only their associated support groups. So tickets that are "Not Assigned" to any group aren't easily seen and typically seem to be forgotten.

For example, Tech A is in the "Network" support group but not in "Applications". If Tech A picks up a ticket from the "Applications" support group it assigns it to Tech A, but marks the Support Group as "Not Assigned".

Could we have an option for when a ticket is picked up by a technician from a Support Group they are not associated with, it will ask which of the Support Groups the technician is associated with to assign to.