This is not an immediate problem, but one that often crops up and never fails to have me wondering. Take this sentence: I went to the supermarket yesterday, or should that be shopping mall, and bought a new coat. Now forgetting the fact that I could compose this differently and solve the problem that way, what do I do about the fact that the 'mid-thought' between the commas is a question and requires the appropriate punctuation mark? A question mark followed by a comma would be wrong, so what would I do if I absolutely insisted on using this sentence as it stands?
The grammar rule is, you don't put the question mark anywhere but at the end of a question. You don't need it for a question within a sentence.
I agree with @GingerCoffee, but will add one more level of complexity that you did not solicit and say that the question imbedded within the sentence is such a "loud" parenthetical interjection that I would have wirtten it with dashes (or m dashes or n dashes or whatever the hell they're called). Important to note that I am not proposing a rule, but only my personal choice. I went to the supermarket yesterday - or should that be shopping mall - and bought a new coat.