They claimed the group sex with Nicholas was consensual and were found not guilty at the 2006 trial, but the jury was unaware Shipton and Schollum were already in prison on other rape charges.
Haumaha was interviewed for Operation Austin and spoke highly of his friends, describing Shipton as a "softie" and Schollum as a "legend" with women.
He also described Nicholas' allegations as "a nonsense" and that "nothing really happened and we have to stick together".
At the committee, Bush repeatedly refused to answer questions about Haumaha, including what he knew of Haumaha's comments and when, or whether the case had undermined confidence in police culture.
National MP Chris Bishop asked if the appointment was a risk to the progress police had made.
"Did you know about those comments when Mr Haumaha was being promoted from assistant to deputy commissioner?
"Do you accept that it is a reasonable response for people to worry about the culture of police with the second in command ... being on the record so publicly having said those remarks?"
Bush replied that he did not want to answer before the inquiry is released, adding his assurance that police will continue to make progress and that "we are now in a really good space".
"I'm as keen as anyone to see the result of that report."
MPs from Labour and National then sparred about whether the commissioner should answer the question.
At one point, National MP Nick Smith said that Labour MPs had blocked a motion to postpone the hearing until after the inquiry was released, saying it was "cynical manipulation by the police and the Government".
Labour MPs said they wanted to call Bush back to the committee after the inquiry was released - if appropriate.
Smith said that Haumaha's comments had prompted Nicholas to doubt how much police culture had improved.
"She has said publicly the appointment to Deputy Commissioner has undermined confidence that police has actually moved forward. Why should we have confidence when Louise Nicholas doesn't?"
Bush said that the public can have confidence in the health of police culture, and he remained committed to ensuring progress.
Bishop then asked about the allegations of bullying against Haumaha in 2016, but Bush said that the Independent Police Conduct Authority was investigating and would release its report soon.
Smith then asked about Haumaha's association with NZ First, but Bush again declined to comment, citing the IPCA investigation.
After the hearing, Bush said he had confidence in Haumaha, who had been working from home while the inquiries were conducted, and that public trust and confidence had improved in recent months.
Bishop criticised the Government for not releasing the report prior to the committee hearing.
"They cynically delayed the release of the report until after the committee finished its hearing this morning and going into a two-week recess. It's a disgrace."
But Police Minister Stuart Nash said it was simply "unfortunate timing".
"It's not a matter of withholding it in order to stymie the committee in any way, shape or form. It's purely a matter of timing, that's all."
The Scholtens inquiry was announced after the Herald revealed in June controversial comments made by Haumaha during the Operation Austin.
It also considered allegations of bullying against Haumaha in 2016, when three women walked out of Police National Headquarters and refused to return.
Scholtens considered the 2016 allegations in her inquiry, but only in terms of what information was potentially available during the recruitment process which led to Haumaha's appointment in May.
The IPCA inquiry is now investigating formal complaints laid by two of the women, and is expected to report back at the end of November.
