How do psychologists ensure they are measuring innate intelligence and not just education? Certain groups perform worse on IQ tests than other. How do psychologists ensure that those differences aren't purely related to differing levels of education between those groups?

How do we know that IQ is a real phenomenon, and not just a purely social factor based on level of intelligence?
Short answer
Any intelligence test will be designed for a certain target population. That target population's score is set at an index of 100 for the standard IQ test. The same test applied to people from a different cultural background, or other socioeconomic status may yield numbers that are unreliable, as they my not understand the question (albeit knowing the answer), or they may lack the vocabulary, or even interest altogether.  

Background
IQ is highly dependent on education; For instance, an illiterate person cannot perform an IQ test at all. And indeed, education can generally affect IQ scores (e.g., Dear &amp; Johnson, 2010). Further, IQ tests are Western constructs and do not apply in many other cultures. So IQ tests are anything but objective. They only apply validly to compare people with a certain educational background. Bush people in the outbacks of Africa may not use numbers at all, but instead may rely on 'one', 'two', perhaps' three' and 'many'. They may be highly intelligent, we simply cannot assign a number to it, simply because they don't care about numbers, geometry, classifications, logic, and other standard components of IQ tests. The APA says rightfully 


  Researchers of cultural differences in intelligence face a major challenge, however: balancing the desire to compare people from various cultures according to a standard measure with the need to assess people in the light of their own values and concepts.


Reference
- Dear &amp; Johnson, Int J Epidemiol (2010); 39(1): 1362–9