Questions don't automatically belong within quotation marks. If it's a question that someone else has asked (or otherwise a quotation from someone or somewhere) then it belongs in quotation marks. But if it is a question that you are posing, it doesn't need quotation marks. You can simply write:
The question is, how does the race question fit in as an overlay to all of this?
The question is: how does the race question fit in as an overlay to all of this?
Note: in American English there is normally a capital letter after the colon ("How"); in British English, lower case is usual.
Whether "race question" belongs in quotation marks is a personal choice. Leaving the quotes off suggests that you know what the race question is and you think that "race question" is a good name for it; adding the quotes suggests that "race question" is someone else's term or that you aren't quite sure that it's the appropriate term.
If you decide to enclose the question within quotation marks and if you also decide to enclose "race question" in quotes, it's true that if the question as a whole is enclosed in double quotes, "race question" should then be single quotes, whereas if the question as a whole is in single quotes, "race question" should be in double quotes.
It's often said that outer quotes are double in American English and single in British English. Actually, British English style varies: novels tend to use single quotes, while most newspapers and magazines use double quotes. Either way, if you use single quotes for the outer quotation, you use double for the inner - and vice versa.