The GSS asked respondents how warmly they feel about: 1) whites, 2) blacks, 3) Hispanics, and 4) Asians. Answers ranged from very cool to very warm. I correlated answers for whites. The mean correlation is .63. While there is substantial range--from .45 for white-white to .81 for Asian-Hispanic--the main story here is that if a person dislikes one group, he tends to have comparatively cool feelings for all groups, even his own. I found the same basic results when I limited the analysis to the black sample or a sample of those of some other race.
We can construct a typology here for all races. Roughly half feel warmly toward all races. Thirty percent or so are neutral toward everyone. Fifteen percent like their group but dislike everyone else, and five percent can't stand anybody. The consistent attitude for various groups leads me to think one's disposition toward groups is a personality trait. It's probably related to one's attitude toward people in general: trusting and warm versus suspicious and hostile.