St Peter's Trust Board chair John Erkkila today told the Herald the investigation by former chief judge of the Employment Court Graeme Colgan would continue with Colgan expected to report back to the board next month.
"He will be reviewing various policies and procedures and will provide recommendations for improvements or changes," Erkkila said.
He said the investigator's recommendations regarding his assessment of various policies and procedures would be provided to the school community "in due course".
WorkSafe this week told the Herald more complaints had been made about bullying at the school since media coverage earlier in the month.
A spokesman said the organisation would continue to work with the school on "staff wellbeing" and gather information despite the resignation.
The workplace health and safety regulator had not opened its on investigation at this stage but findings from the school's investigation would help determine any future actions.
Erkkila said staff were told of the resignation at a staff briefing yesterday morning and parents were informed by email immediately after. Students were told during an assembly the same morning.
Williams remained absent from school.
In an earlier email to parents Erkkila said if it was "deemed that workplace bullying has occurred" it would be addressed.
"I cannot and will not allow bullying of any kind in our workplace. Being on the receiving end of bullying behaviour is incredibly damaging to a person's mental health and it will not be tolerated, now or ever.
"If there has been or if there is still workplace bullying happening, we need to know so it can be stopped."
Operational matters remained in the hands of chief operating officer Rob Campbell while teaching and learning matters would be looked after by campus principal Julie Small until a new principal had been appointed, Erkkila said.
"Although early days, the board has considerations it needs to make before any hiring process gets underway," he said.
Burden joined the elite Waikato school in 2016 after a decade as headmaster of Mt Albert Grammar School in Auckland.