Security agencies are under pressure to explain why they appear to treat Muslims as more of a threat than the alt-right and white supremacists.
The inquiry will look at what the agencies could or should have known about the man accused of Friday's Christchurch terror attacks and any impediments to sharing information, Jacinda Ardern said.
The Islamic Women's Council has said it told the Department of the Prime Minister and Cabinet at a January 2017 meeting of the 'extreme urgency' of its concerns about rising racism and the alt-right, and also told this to the SIS.
Former Race Relations Commissioner Dame Susan Devoy said the response of officials to Muslims over what they see as a growing threat to them was "diabolical".
Dame Susan was instrumental as Race Relations Commissioner in helping Muslims alert authorities to their security fears.
She said trying to get public officials to meet them was incredibly difficult, and even when they got in the door, no action was taken and they got little support.
"I'd be asking our government, what is our countering violent extremism strategy? Because one of the pillars of that is to engage with your community. And I would think that we've failed miserably on that count and should feel very ashamed of ourselves. I can assure you, it wasn't just the Muslim community, the Jewish community also felt at threat."
Dame Susan said State Services Commissioner Peter Hughes was one top official that Muslims met with.
Mr Hughes in a statement said his commission's role is to ensure the right agencies help and that various agencies have been working with Muslim groups on social inclusion.
A Wellington man, Asher Wilson-Goldman, tracked and disrupted far-right online collaboration with Australia for five years and said his group believed the state agencies had no appetite to surveil the far-right.
He said he kept police informed, and in 2005 helped them identify far-right people from photos, however during the same meeting, police produced pictures of conservationists protesting a road bypass and pressed him to name them.
He said the state was prioritising surveillance of one over the other even though the proven threat in New Zealand is all one way.
"There's a long history in New Zealand of fascists and neo-Nazis, whether organised groups or individuals, undertaking attacks. Beatings with basebell bats, smashing up on mosques, graffiti attacks on synagogues, and the police and the intelligence service have ignored it in favour of surveilling the very people who were attacked on Friday."
Responding to the criticism this morning Labour MP Andrew Little, the minister responsible for the two spy agencies, denied they had taken their eye off the threat posed by right-wing extremists.
"Under the present leadership of the two spy agencies, the culture of those organisations has improved considerably but I think we need to have the inquiry to know and understand whether there still remain blind spots and whether we can improve further on what is there at the moment," he told Morning Report.
Mr Little said the inquiry would help determine if the country's state agencies had any blind spots "to see whether the nature of the organisation meant that these weren't being observed when they might've been in plain sight".
"Some of the warrants deal with the threat of extremism, and the warrants that deal with the threat of extremism are not confined to one form of extremism.
"I'm satisfied from what I've observed that the SIS in particular is looking at all forms of extremism, as the prime minister has said, for the last nine months they've been reviewing specifically the rise of the alt-right and white supremacy and that's been an explicit part of their programme."
"Every tipoff relating to right-wing extremism and white supremacism has been followed up."
Mr Little noted there was no set timeframe for the inquiry, which he said could be an interim one followed by a more thorough investigation.
The censored security agency briefing to the incoming minister in 2017 said the terrorism environment was dominated by Islamic fundamentalists and "violent extremist ideology" and online messaging was resonating with some people - but it did not mention the far right.
Prime Minister Ardern said yesterday that security agencies took white supremacists seriously.
The Five Eyes partners intelligence agencies' increased focus on the alt-right was the trigger for New Zealand's ramp up in observing white supremacy more closely in the past nine months too, Mr Little said.
Both the SIS and GCSB said they had no intelligence themselves or from any of their partners about the Australian man who has been charged over the shootings.
Social media tracking of comments and individuals
The man accused of Friday's terror attacks in Christchurch was also active on social media in expressing his views.
Dr Coyne said it was hard to distinguish between an immature teenager's ranting and a committed ideologue's rant among a massive amount of data.
"The expectation that they can track everybody at the same time is very, very much ill-informed."
Dr Coyne said that in Australia there was a list of about 300-400 people that were being managed as the highest threats.
"It's impossible to do physical surveillance on them all the time, let alone watch all of their activities online."
He said security agencies on both sides of the Tasman would have been working feverishly since Friday to find out more about the accused gunman and if he should have been on a security watch list.
"The places that people hide that kind of information, the way that the remain anonymous, those things are consistent over the two different types of offence.
The difference was that investigation of paedophile offences starts from a known illegal image, and from there people are identified.
"When you're talking about extremists there isn't that clarity of who has created a breach of the law to start identifying them."
Mr Little said the inquiry would deal with whether anything online should have raised red flags.
"The advice I've had is that the extent that [the accused] made comments online they were closed Facebook groups that unless you know they're there they're not readily accessible.
"I think we've got to be clear as well the way our agencies operate, they don't monitor every digital activity that happens at any time of the day, what they do is based on an assessment of national security threats and risks and based on what agents themselves observed."
He said New Zealand and other countries were considering what could be done to get social media platforms to remove hate speech and videos.
"There are a lot of people who say things on social media that are kind of big-noting. They can be vile, they can be offensive but there's no either means or intention to carry through the comments that are made through expressions of violence.
"On the other hand, there are people who make violent threats towards classes of people, groups of people, types of people.
"The role of the agencies, among other things, is to do their best to distinguish between what is a real threat and what is just big-noting on social media."
This article was first published on Radio NZ.