In response to Ardern's remarks, Morrison said the woman's Australian citizenship was revoked automatically and that it was his job to put his country's national security interests first.
"They left New Zealand at the age of six, were resident in Australia from that time, became an Australian citizen, left from Australia to Syria, and travelled on an Australian passport", she said.
In an unusually blunt message to her counterpart Scott Morrison, Ardern said Canberra was "wrong" to expect New Zealand to accept the woman, who she said had strong ties to Australia.
"New Zealand, frankly, is exhausted of having Australia exporting its problems", Ms Ardern said.
He points to New Zealand's reluctance to take responsibility for up to a dozen Kiwi jihadists in the Middle East, including Mark Taylor, who is being held by a Kurdish organisation.
The woman is facing deportation to New Zealand after being caught entering Turkey from Syria with two children.
One New Zealand man is known to be detained in northeast Syria. Instead, Australia revoked the woman's citizenship, making her exclusively New Zealand's problem.
Janidra Ardern, New Zealand's Prime Minister, Foreign Minister Nanaia Mahuta, and Defense Minister Peeni Henare announced that the country will withdraw its troops and end the military missions in Afghanistan.
"We do not want to see terrorists who fought with terrorism organisations enjoying privileges of citizenship, which I think they forfeit the second they engage as an enemy of our country", he said during a press conference in Canberra.
Mr Morrison said he would speak to Ms Ardern later in the day.
"The welfare of the children also needs to be at the forefront in this situation". "These children were born in a conflict zone through no fault of their own", Ms Ardern.
It is needed not just for case like this, but for other types of extremism already visible in New Zealand. "They did not act in good faith", she continued. "We know that young children thrive best when surrounded by people who love them", she said.
About 1,500 Kiwis have been caught in the dragnet - nearly half the total of 4,000 - and Wellington has raised concerns some of those deported allegedly grew up in Australia and have no ties to New Zealand.
Since 2014, around 3,000 New Zealanders in Australia have had their visas cancelled "on character grounds" - which does not always require a criminal conviction.