But she was named in the New York Times today - and subsequently in New Zealand media - with confirmation from the International Committee of the Red Cross (ICRC), which issued a plea for any information that could lead to her location.
Islamic State territory was wiped out with the fall of Baghouz last month, but security forces have not yet been able to find Akavi or get confirmation of whether she was still alive.
The Government is continuing to work on the basis that Akavi is still alive, but disagreed with the decision to name Akavi publicly.
"It absolutely remains the Government's view that it would be preferable if this case were not in the public domain," Ardern told media at her post-Cabinet press conference today.
"For that reason I won't be commenting further on it, with one exception and that is to make special mention of the domestic New Zealand media.
"The decisions that have been made over a period of time by various outlets and journalists has not only been responsible, I think it's been exemplary. I'm sure I speak on behalf of successive governments when I say 'thank you'."
She did not explicitly say that naming Akavi could endanger her life, but it has been the long-standing Government position and she said that position had not changed.
Ardern repeatedly batted away questions including whether naming Akavi was irresponsible, whether Akavi was still alive, whether the Government had considered paying a ransom for Akavi, or whether the Government's non-combat unit in Iraq that was looking for her was still based there.
She said the Government had no plans to change its policy of not paying any ransoms, which is seen as potentially encouraging more hostage-taking.
Asked if it was reckless for the ICRC to name Akavi, Ardern said: "Those are decisions for them."
The Government had told the ICRC of its position, she said.
"Decisions were taken that were not our own, and I won't be commenting any further on decisions made by others.
"I'm not commenting or speculating on this case in any regard ... The Government's position has not changed. We prefer this not to be in the public domain."
In the days after the terrorist attack in Christchurch, IS spokesman Abu Hassan al-Muhajir called for revenge attacks - but Ardern would not say if this was a factor in her decision not to comment.
She said she believed the previous Government would also have been reluctant to comment.
Earlier Foreign Minister Winston Peters said efforts to locate Akavi were continuing.
"Where a New Zealander is held by a terrorist organisation the Government takes all appropriate action to recover them. That is exactly what we have done here."
The last credible sighting of her was in one of the last Islamic State strongholds in eastern Syria in December.
Another possible sighting of her more recently at the Al-Hawl refugee camp in north-western Syria, near the Syria-Iraq border, turned out to be wrong.
Intelligence officials told the Government on Wednesday last week that the case was under review, meaning they needed to reassess what might have happened to her. Being under review does not mean that she is thought to be deceased.
The following day, April 11, the family had one of their regular meetings with Ministry of Foreign Affairs and Trade (Mfat) and Red Cross officials.
This was followed by a meeting at the Beehive with Peters and an Mfat official.
It was the first time Peters and the family met face to face.
It is unclear whether the family supported Akavi's becoming public, but they understood the ICRC was working, and had been working, in Akavi's best interests.
Peters offered the family any support they needed.
At the end of the meeting, Peters is understood to have told the family to keep praying, and that he hoped everyone's prayers will be answered and she will be found alive.
