'Volte-face' means 'change of opinion'. Thus, from the sentence we can conclude that the correct phrase should mean 'changed their opinion'.
Option A is incorrect. 'Did a volte-face' is not a correct phrase and thus, it is incorrect.
Option A is incorrect. 'Carried on a volte-face' is nearly same as the given incorrect idiom. Thus, it is also incorrect.
Option A is incorrect. 'Cut a volte-face' is also an incorrect idiom. Hence, it is not the answer.
Option A is correct. 'Made a volte-face' means 'to change ones opinion'. Thus, it is very close to the required meaning and so, it is correct.