The Commission or Appointing Authority may require medical and/or psychological examinations of an employee, class of employees or prospective employees for the purpose of determining fitness for the position held or to be held. Such medical and/or psychological examination shall be made by a physician, or a psychologist for a psychological examination, designated by the Commission and the results recorded and filed with the Commission. Employees who either refuse to submit to such medical and/or psychological examinations when so ordered, or found to be unfit for the duties of their positions after such psychological or medical examinations are subject to be suspended, transferred, demoted or dismissed as the facts may warrant after charges have been preferred and hearings held. Any employee suspended, demoted or dismissed under the provisions of this section shall have the right to appeal to the Commission.