For an overview of the Liberty Party in the Taylor/Fillmore administration, see the first convention
of the four held by the Liberty Party in 1852.
The third Liberty Party convention of 1852 encompassed the seceders from the second convention.
When the second convention passed a resolution sponsored by Gerrit Smith to form a union with the Free Soil Party, the minority faction bolted. The New York Times article on the seceders' convention has several inconsistencies, including calling the bolters the majority faction.
The bolters met in the Free Church in Syracuse NY at 5:30 p.m. on 9/3/1852. They adopted the "majority" report from the second convention (which had been rejected).
The convention then nominated William Goodell for President and Charles C. Foote for Vice President.
It appears that Foote declined to run. Goodell continued his campaign, but at the fourth convention of the party for 1852, a different candidate was chosen to run for VP.
Popular Vote of 1852