All officers holding office under the previous Charter, except the members of the Common Council and School Inspectors holding office under and by virtue of such previous Charter, shall continue in office until their successors shall be chosen and shall qualify, or until the office is abolished in accordance with the provisions of this Charter. Where questions arise concerning the responsibility and authority of officers, pending the appointment of a City Manager, the Council shall temporarily fix such responsibility and grant such authority as is necessary to maintain the services of the City.