[With regard to any county which is exempt from the provisions of Section 19-2-3, this section shall read as follows:]
The board of supervisors of any county in the state may construct, reconstruct, maintain, or contribute to the construction, reconstruction, and maintenance of any state highway declared by legislative act expressly to be such highway or which, in accordance with law, has been taken over by the State Highway Department for construction, reconstruction, or maintenance; and this section shall apply to such state highway within as well as without the limits of any municipality and, when within such limits, with or without the consent of the municipal authorities. In such cases and for said purposes the board of supervisors, for and in the name of the county, may fully exercise the right of eminent domain within any municipality, and it shall be the duty of the board so to do on the lawful demand of the State Highway Department. This section shall apply also to said state highways within a separate road district, or districts, adjacent to a municipality.