In cities, the treasurer, the counselor, the clerk or register, and the comptroller, where such office exists, the chief officer of the fire department, four members of the fire department and one of the retired or pensioned members, whose term of office shall be for one year and who shall be appointed by the mayor on or about the first Monday of December of each year, shall constitute and be a board to be known as “The Board of Trustees of the Firemen’s Pension Fund”. The board shall select from their members a president and secretary. The members of the fire department, and the retired or pensioned members, shall each hold an election on the first Monday of December of each year and respectively select four members of the department and one retired or pensioned member for recommendation to the mayor for appointment as members of said board, but the mayor shall not be required to appoint the candidates so recommended.