Section 56. For Suffolk county, the duties prescribed for the board by section fifty-one, except clause (c) thereof, shall be performed by the city council of the city of Boston, subject to the provisions of the charter of said city; the duties prescribed by sections forty-nine to fifty-five, inclusive, for the director of accounts shall be performed by the budget commissioner of said city, and all other duties prescribed and powers conferred by said sections forty-nine to fifty-five shall be performed by the mayor of said city. Any other county is hereby authorized, subject in all respects to the approval of the governor and council, to establish agencies for the purpose of establishing and administering classification and compensation plans in accordance with the provisions of sections forty-nine to fifty-five, inclusive, and agencies so established shall have the powers granted by said sections to the board and the director of accounts. Upon the establishment of such agency, such county shall cease to be represented in the board or to participate in the election of its members, and its obligation to contribute toward the expenses incurred by the board and by the director under section fifty shall terminate.