Of course they support him. He's one of them in sheep's clothing.
There was a time when the CPUSA ran its own candidates for reasons of Soviet prestige. Now with the Soviets gone, the CPUSA are mere hangers-on of the "movement" and really aren't important as an organization.
For those who are more "radical" than the average liberal, there are newer outfits like ANSWER that lead the way ideologically.
Red sKerr 2004.