According to the General Handbook of Instructions it says that a stake president must set apart an Elders Quorum president because of the keys related to his calling as president. Is there a situation where a counselor in the stake presidency can do this setting apart? And if so, how are the priesthood keys passed down?
As you have indicated, the General Handbook of Instructions requires Elders Quorum presidents to be set apart by their stake presidents, and you also cited the reason for this requirement. Counselors to the stake president may set apart counselors to the Elders Quorum presidents, and they may also set apart High Priest Group leaders and their assistants because no transfer of keys of authority is involved. I know of no exceptions to this procedure.