The supervising officer may transfer patients of the state hospital or life skills and transition center between such institutions whenever the superintendent of any such institution shall recommend such transfer and the supervising department is satisfied, upon investigation, that such transfer is advisable. If any patient, so transferred, is maintained at the expense of the county from which the patient was committed, the cost of the patient’s maintenance in the institution to which the patient is transferred must be charged to such county and must be collected therefrom upon notice to the county auditor of such county by the supervising officer.