GSS respondents were asked to rate the intelligence of various groups, giving them a score somewhere between untelligent (1) and intelligent (7). The mean ratings given by blacks and whites are shown above (sample size = 1,230).
Blacks and whites agree that Jews are the smartest. It's reassuring to see that people aren't completely clueless about reality. Blacks think Asians are smarter than whites, but whites think that they are a bit smarter. Blacks view themselves as being almost as smart of whites; whites disagree. Whites consider Hispanics to be less intelligent than blacks--that surprises me--it's possible that the fear of looking anti-black is being reflected here. The fact that whites don't give southern whites a score much higher than blacks might reveal the ease with which so many whites look down on their brothers in the South. Blacks also think they are smarter than southern whites.
So, whites overestimate their IQs relative to Asians, while blacks do so basically with everyone. They do think whites are smarter, but by only a trivial amount.