The Absolutists were the "old guard" of Andorian society, who were loyal to King Collev and his family. Following the death of Collev and his heir, Bregev, many within the Absolutists favored Levak, but were persuaded by Federation negotiator Jean-Luc Picard to strike a deal with the rival group Red Sash and support former King Thurl.
A small group within the Absolutists, though, assassinated Thurl before he could return from exile, and planned to install Levak. Thurl's son, Yevan, was able to stop this plot, and claim the throne himself. (TNG - Enterprise Logs short story: "The Captain and the King")