Footage of the auction was shared on Youtube on Sunday night and pictures of the event were also posted by teenagers on the photo sharing app, Instagram.
The mosque says the proceeds from the sale were for the mosque's benefit, claiming the Shahada symbol that features on the flag had been "hijacked" by Islamic extremists.
It is believed the flag was sold for more than $2000.
New South Wales Premier Mike Baird said the government would be taking action against anyone supporting ISIS, which is listed as a terrorist group by the federal government.
"We expect everyone in New South Wales to obey the law or face the consequences," he told the Daily Telegraph. "All parents and all communities need to protect young people from the insidious and corrosive effects of the radical ideologies that are causing so much suffering around the world."
New South Wales Police said they are unable to make arrests over the auction as the funds raised are for the mosque.
