In each county there must be established an absentee voting precinct located in the office of the county board of voter registration and elections. The county board of voter registration and elections, municipal election commission, county committee for each political party, or executive committee of each municipal party in the case of primary elections is responsible for the tabulation and reporting of ballots at the absentee voting precinct. The absentee ballots must remain in the custody of the county board of voter registration and elections until transferred to the county board of voter registration and elections, municipal election commission, county committee for each political party or executive committee of each municipal party for the purpose of tabulation and reporting as provided in § 7-15-420.