To aid in administering its permit system, a district board may contract with any county or city included, in whole or in part, within the district, and any such county or city may contract with the district, for the performance of such work in the name of, and subject to the approval of, the district air pollution control officer by the building department or other officer, department, or agency of the county or such city charged with the enforcement of regulations pertaining to the erection, construction, reconstruction, movement, conversion, alteration, or enlargement of buildings or structures.