The governing body of any incorporated city that has adopted the home rule provisions of chapter 40-05.1 and the department may enter into contractual agreements under which the department may collect any motor vehicle registration fees assessed by the city. Agreements entered into under this section must provide for an agreed-upon amount to be allowed the department for services rendered in connection with the collections. Any agreed-upon amount collected must be deposited in the motor vehicle registration fund. The department shall deposit with the state treasurer all money collected under this section and shall accompany each remittance with a certificate showing the city for which it was collected. The state treasurer, quarterly, shall pay to the city auditors of the several cities the money to which they are entitled under this section.