Where, at the time of any annexation or merger as provided in sections 14-117 to 14-125, the municipal license year, for any kind of license, of any city or village annexed or merged with a city of the metropolitan class as provided in such sections extends beyond or overlaps the municipal license year of the city of the metropolitan class, then the proper authorities of the city of the metropolitan class may issue to the lawful holder of any yearly license issued by any such city or village annexed or merged with the city of the metropolitan class, or to any new applicants applying for license to continue the business at the place covered by such expiring city or village license, a new license under such conditions as may be provided in the laws or ordinances governing the city of the metropolitan class for the remainder of the city of the metropolitan class license year, extending from the expiration of such city or village license up to the end of the city of the metropolitan class license year, and charging and collecting for such license only such portion of the yearly amount fixed for such license by the laws or ordinances governing the city of the metropolitan class as will represent proportionately the time for which the new license shall be granted.