An auxiliary verb is a verb that is used along with another main verb to form tenses, negatives, and questions. Examples are: 'be', 'have' and 'do'. It usually comes before the main verb.
The auxiliary verb, 'have' is used to form perfect tenses or used with an infinitive to express an obligation (Example: I have to be there). In the given sentence, the clause, 'They _____ not want to go' is in simple present tense form and not in the perfect tense. The main verb is 'want' and it is not in infinitive form. There is no mention of obligation either. Hence, the auxiliary verb, 'have' cannot be used. The words, 'have' and 'has' are forms of the auxiliary verb, 'have'. Hence, both options C and D are incorrect.
The auxiliary verb, 'do' is used to form negative of the simple present tense. In the given sentence, we see the word, 'not'. This indicates that the simple present tense of 'want to go' is getting converted to negative form. Hence, the correct answer is either 'do' or 'does'.
If the subject is in the third person, singular form, the auxiliary verb, 'does' is used. Otherwise, 'do' is used.The subject, 'they' is in the third person, plural form. Hence, option A is correct and option B is incorrect.