The state registrar shall appoint the chief administrative officer of each county health department as the county registrar. All persons in the county required by law to file reports of birth, death, and fetal death shall forward these reports to the county registrar. The county registrar may appoint a deputy registrar who is vested with the right to carry on the duties of the office. The county registrar and deputy registrar shall carry out the duties formerly carried out by local registrars without additional compensation. The county registrar shall appoint a subregistrar for each hospital, nursing home, and other institution as required within the county whose duty it is to issue Burial-Removal-Transit Permits for deaths occurring at the hospitals, nursing homes, and other institutions. The county registrar shall require the coroner of the county to issue Burial-Removal-Transit Permits for deaths occurring outside hospitals, nursing homes, or other institutions.