If the headmaster had told the child “I am punishing you because your parents have asked me to do so” then I think it would be accepted that he was acting on their behalf, rather than on behalf of the local authority. This could be the case even if the offense was a school offense because a parent can punish a child for a school offense (secondhand, if you like, for not obeying school rules). He would be breaking local authority rules by doing this in school time and on school premises, but that is a civil matter.

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