Whenever a special tax shall have been levied under the provisions of this Division of this Code, the highway commissioner of any road district may, by agreement with the corporate authorities of any municipality of less than 20,000 population, extend any road improved under the provisions of this Division of this Code within or through the corporate limits of such municipality. The provisions of this Section shall also apply to any municipality of more than 20,000 population when a portion of such municipality lies within the road district and the population of such portion does not exceed 20,000. However, such extension within such municipality shall be of the same cost and kind of material as the road outside such municipality, to be paid for out of such special tax and after completion to be maintained by the corporate authorities of such municipality at the cost of such municipality.