In this case, the word is being used as a noun. Stavropoulos gives just one example of the noun use: ο υπεύθυνος ενός πανδοχείου νεότητας: warden. Collins dictionary gives head (τμήματος πωλήσεων, λογιστηρίου), manager (καταστήματος, τροφοδοσίας, εργαστηρίου), and renders ποιος είναι ο υπεύθυνος εδώ as who is in charge here?
From this it will be clear that there is no single English equivalent, and you will have to think who would be the person or people in a school who would make the kind of decision referred to in the text. It would be normal in English to refer to them by the title of their position, since this would be known to the mother who is relating the story.