"Yesterday I travelled to Sydney with Emirates. I have never experienced service like it in all my life. It was fantastic, both on the ground and in the air. Amazing."

He said his ticket was not a freebie and responded to questions from followers that he was undergoing "very strict 14-day quarantine" upon his arrival although did not say where. In August, he used Twitter to seek help from British Airways in getting to Australia.

Melbourne man Julian, who preferred to use just his first name, was in Britain in July on business when the government announced the cap restricting the number of arrivals in a single week to about 4000.

The 52-year-old said he held gold frequent flyer status with Emirates but had been bumped off six flights as a result of the 30-person limit on a single flight.

"COVID is proving one rule for one and one for another every single day," he told The Sydney Morning Herald and The Age.

"I find it a bit rich that Emirates would fly Lord Sugar, a non-Australian, who hasn’t flown commercially for 25 years, so never with Emirates, when I am a Gold Member, have spent thousands of dollars to get that, and yet cannot get back from London.

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"What 'essential business' is Lord Sugar travelling on? A TV show appearance?"

Diane Lee, 57, runs a Facebook support group for Australians stranded overseas by the caps.

She has finally made it home from Vietnam and is serving out her hotel quarantine in Sydney.

"I'm angry on behalf of other people who have been abandoned by the Australian government who give preferential treatment to rich celebrities," she said.

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"How is this fair?"

"Australian families — who have given up jobs and homes to catch flights home — are living in airports and in homeless shelters because their flights are continually being cancelled.

"The Australian government should be ashamed that it has abandoned its own citizens during a global pandemic, and welcomed elites with open arms."

Emirates, Sugar and Home Affairs Minister Peter Dutton have been contacted for comment.

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Separately, Amnesty International said the caps were a violation of Australians' human rights.

"Several articles of international law explicitly state that people have the right to leave and return to their own country and not be deprived of the right to do either," Amnesty said in a statement. It backed calls to increase quarantine capacity by using federal quarantine facilities.

The federal government banned foreigners from entering the country in March but exemptions can be obtained by applying to the Border Force Commissioner.

The national cabinet agreed in July to cap the number of people who can enter the country to about 4000 a week, after breaches of the quarantine system in Victoria triggered the state's second outbreak.

The cap has created a queue of at least 30,000 Australians around the world who have told the government they wish to return home. Airlines say the figure is far higher and is about 30,000 in Britain alone.

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Prime Minister Scott Morrison has ordered the states to accept 2000 more people from overseas into quarantine hotels each week ahead of a national cabinet meeting on Friday.

Carriers are offering only first or business class tickets to try to recover costs for each flight and some have stopped selling tickets to Australia altogether.

Passengers have repeatedly complained of being bumped from flights at the last minute for those paying higher fares or other priority.

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Opposition foreign affairs spokesperson Senator Penny Wong questioned the government's priorities in light of the exception for Sugar.

"Why does Lord Sugar get first-class treatment from this government, while tens of thousands of Australians get left behind?

"Once again we see with Scott Morrison it's one rule for the privileged few while others are told to ask for charity or go a homeless shelter because they can't get home."

A Border Force spokesperson said the department could not comment on individual cases but "travel exemptions to Australia may be granted under a number of categories, including persons delivering services in sectors critical to Australia's economic recovery (such as financial technology, large scale manufacturing, film, media and television production and emerging technology), where no Australian worker is available". The person said any support from a State, Territory or Commonwealth government authority would be considered favourably by decision makers.

Sugar's case is the latest in a string of high-profile exemptions granted to sporting figures and celebrities who have been able to travel in and out of Australia despite the bans and caps causing chaos and misery for tens of thousands of citizens and their families.

Last month, former prime minister Tony Abbott received an exemption to travel to Britain where he spoke to right-wing think tanks, attended a golf tournament in Wales and dined with his new boss, British Trade Secretary Liz Truss.