The man charged with murdering Constable Matthew Hunt in a West Auckland street last month will keep his name secret for another two weeks.

His co-accused has again been refused name suppression for a second time - and will be named later today if she does not instruct her lawyer to mount an appeal against the decision.

Hunt was allegedly murdered during a routine traffic stop on Reynella Drive in Massey on June 19.

His funeral was held last week at Eden Park.

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A 24-year-old man is charged with murdering Hunt, attempting to murder another police officer and wounding a member of the public as he allegedly fled the scene in a car.

A 30-year-old woman is charged with being an accessory to murder after the fact.

It is alleged she obtained and drove the getaway car for the accused shooter after Hunt and his colleague were gunned down.

Both have pleaded not guilty to the charges.

Neither has been identified at this stage due to suppression orders.

The man was granted interim suppression when he first appeared in the Waitakere District Court, on the basis that identifying him at that stage could hinder the police investigation and could affect his fair trial rights.

The woman was refused suppression at her first appearance.

She then appealed that decision, meaning suppression remained in place until she could argue her case in the High Court.

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Justice Venning heard both applications for continued name suppression in the High Court at Auckland this morning.

The details of the arguments and other issues discussed in the hearing cannot be published.

But Justice Venning allowed media to publish the outcome.

Constable Matthew Hunt. Photo / Supplied
Constable Matthew Hunt. Photo / Supplied

He extended name suppression for the alleged shooter for two weeks, where a final hearing will be held and a decision made.

But he refused to continue suppression for the woman.

She was not in court today and her lawyer sought a delay on her name being published so he could take instruction on whether she wanted to mount a second appeal.

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If she does not appeal - her name will be published upon Justice Venning's direction.

If she does appeal, name suppression will continue until that can be heard.