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Rennie, a New Zealander, took over Australia's management in July 2020 from the Glasgow Warriors.
Australia coach Dave Rennie said that he was "bloody mad" at New Zealand for cancelling their Bledisloe Cup match, without informing the Wallabies.
Covid-19 restrictions prevent New Zealand from traveling to Perth for the 28 August fixture.
After the country's first positive Covid instance in six months, New Zealand announced a snap lockdown.
"I'm bloody angry. Rennie said that all of our boys found out about each other through social media.
"I thought that 'New Zealand Rugby didn’t even have respect to consult Rugby Australia about its decision'. That's very disappointing.
"I feel like only one of us is doing the right thing for the game's best interests.
"Last year we traveled to New Zealand to make sure the games were played in the best interests of both countries.
"We were quarantined in Wellington for a period of two weeks. We had hoped that they would honor our promise to visit this year. This tells you a lot more about them than it does us, I think."
Two matches were played by the Wallabies in New Zealand, a 16-16 draw as well as a defeat of 27-7.
New Zealand has also announced that the fixtures against South Africa, world champions on 25 September and 2 Oct 2010, have been cancelled. The women's matches with Australia have also been cancelled.
The Black Ferns were to face Australia in Christchurch, 26 September, and Eden Park, 2 October.
"It's extremely disappointing to be informed about this decision via media, even having a conversation only moments prior and there being no mention of this being the intention," said Andy Marinos, Rugby Australia chief.
"We engaged in positive conversations with both the Government partners as well as the teams and felt confident that we were on our way to finding a solution."
New Zealand Rugby (NZR), in a statement, stated that the decision was taken after consulting relevant parties.
It stated that it was following advice from Government that the Springboks or Wallaroos wouldn't be able enter New Zealand to participate in the matches because of Covid-19 travel restrictions.
Mark Robinson, NZR chief executive, said: "We share our fans' disappointment, players, and other stakeholders, especially those in Dunedin, where the sold out 100th Test between South Africa and the All Blacks was shaping up to be a very special occasion.
"However, after discussing with Government, it became clear that there was no way to bring the South African or Wallaroos squads into New Zealand. We understand the importance and need to put the safety and health of New Zealanders first."
New Zealand's men opened the Rugby Championship on Saturday with a record win against Australia to claim a 19th consecutive Bledisloe cup win.
New Zealand was originally scheduled to host the women’s Rugby World Cup in Oct before it was moved one year later to March 2022.