It is perfectly OK to use a question mark followed by a colon. Writers are often egocentric and resistant to constructive criticism, but speaking as an English major and seasoned editor, it is NOT acceptable to finish a chapter with both a question mark and a semi-colon. Although they are both punctuation marks used to form questions, only a question mark indicates that the statement it follows is a question. A semi-colon does not make this clear; instead, it tells the reader that something else needs to be inserted into the sentence before it ends.
A colon should be used only after a full idea; it acts as a traffic signal, indicating that what follows is a definition, enlargement, or explanation. The question in question is none of these—just it's a question, and using a colon (or a comma) before it suggests that it's the only question that exists. This would be incorrect.
"Semicolon and colon"
Mark the question
Towards the conclusion of a phrase The exclamation point is a terminal punctuation mark. As a result, no period or question mark should be used after it. For an exclamatory inquiry, some authors will use both a question mark and an exclamation point, although only the exclamation point is genuinely essential. This combination is called a flangue.
When the second sentence explains or demonstrates anything in the first statement, use a colon instead of a semicolon or a period between the two sentences. If a colon is followed by two or more sentences, capitalize the first word of each sentence that follows. A list of things to include after a colon is called a "colon clause."
In this example, the colon separates two ideas connected by “and”, but not necessarily by sentence structure. The ideas are unrelated and could have been separated by a comma. However, because they are part of one thought, we use a semi-colon instead.
A colon can also be used when giving directions. For example, if you want someone to go north on Main Street and then turn right at Third Avenue, you would say: "Go north on Main Street, then turn right at Third Avenue."
In conclusion, a colon is a mark used to separate ideas in a sentence or clause. Colon clauses help explain how to do something or what items belong together.
How should I format a straight inquiry within a sentence?
There is no hard and fast rule against using two colons, although it is often avoided, mostly because it appears fussy and arrogant. Having said that, I believe your second colon would be preferable as a comma in any case.