As the summer season reaches its height at Royal Ascot next month, one of the sights to behold will be the billboards on the enclosures at this most exclusive of events, sponsored by the gas-rich state of Qatar.

One never quite knows who will accompany Her Majesty the Queen in her traditional ride in coach and horses down the verdant green turf of the racecourse. The odds are that the young Emir Sheikh Tamim bin Hamad bin Khalifa Al-Thani will be among the guests. With its vast wealth, Qatar has established itself as one of Britain's biggest inward investors and its ambitions to own much of London's most prestige real estate knows no bounds.

With a land border with Saudi Arabia and vital strategic locale in a tumultuous part of the Middle East, Qatar is also a pivotal player in the Arab world. It has been a big supporter of the Muslim Brotherhood, the principal movement behind Hamas. It was one of the main funders of the revolutionary groups behind the Arab Spring and has actively supported the medieval jihadists behind Isis (Islamic State). In spite of its funding for such groups, often funnelled through mysterious middlemen, Qatar also poses as a friend of the West. It allowed the Americans a base there during the Iraq war. It hosts the Arab world's only free-ranging news channel, Al Jazeera. Qatar was the only Gulf state to welcome an Israeli representative after the Oslo accords of 1993.

In the past few months it has been rapidly adding to its existing holdings. These include Harrods; the valuable Chelsea Barracks site in West London; One Hyde Park in Knightsbridge; the Olympic Park; and the American embassy site at Grosvenor Square. It owns a 26 per cent stake in grocer J Sainsbury, 15 per cent of the London Stock Exchange, and a 6.7 per cent share in Barclays Bank bought at the height of the 2008 financial crisis.

In January it bought 10 per cent of International Airlines Group, the holding company of British Airways and Iberia. After a legal battle with the Barclay brothers, owners of the Telegraph newspaper titles, Qatar's Constellation Hotels finally bought a 64 per cent stake in three of London's landmark hostelries - Claridges, the Connaught and the Berkeley in Knightsbridge - for £1.5 billion. This added to hotel interests that include the London Park Lane, bought for £300 million in 2013 and the Grand Hotel in Paris, acquired for €300 million last year. Its most recent purchase is a six-storey townhouse in Mayfair this month, for which it paid £40 million. One property expert quoted by London's Evening Standard noted: "The Al-Thani family are looking to build a vast dormitory where you might have an aunt living in one property and a cousin in another."

Just for good measure the Qataris have emerged as buyers of Picasso's $180 million Women of Algiers, sold at Christie's in New York. This, despite the fact it cannot be shown in the Muslim state because of the blatant display of female nudity.

No one can be overly concerned about Qatar overpaying for a Picasso. A bigger worry is its stake in Britain's flag-carrier airline and in some of the most luxury and prestige hotels in London and Paris. Airlines in particular are strategically significant businesses, which is one of the reasons why the Americans do not allow control of their main carriers to fall into overseas hands. At times of tension, there will be concern that Qatar's liking for radical groups across the Middle East could lead to its investment in BA to be abused. Its hotels could also be a place where representatives of terror movements could find refuge.

Qatar's financial integrity is also in question. The decision to pick Qatar to host soccer's World Cup in 2022 has always been controversial, but after the arrest of several officials of football's governing body on alleged corruption charges, that win is looking uncertain.

The US Treasury under-secretary for terrorism noted in last year that Qatar was also giving shelter to two of al-Qaeda's most senior financiers, both of whom are on a blacklist and should be subject to financial sanctions. Moreover, it was no accident that during Israel's longest war, Operation Protective Edge in Gaza last summer, the leader of Hamas, Khaled Meshal, conducted his press conferences from a Qatari hotel room. Under pressure from the anti-Isis leadership in the Middle East, Meshal was expelled from Doha in January of 2015 and is thought to be in Turkey.

Qatar's 34-year-old leader, Sheikh Tamim bin Hamad bin Khalifa Al-Thani, sees nothing strange in his country's embrace of terror movements. "I'm not in a camp against another camp... I have my own way of thinking," he noted in a rare American television interview.

Qatar has hosted the Egyptian and Syrian Muslim Brotherhood as well as Hamas. It regards its relationship with terror organisations as a strategic asset in a dangerous neighbourhood. It has, for instance, embraced the Palestinian idea that Jerusalem should be the capital of an emergent Palestinian state and offered billions of funding to support the cause. Israel and the Jewish community cannot regard the heavy investment of such a quixotic and often hostile state in the UK with any equanimity.

Alex Brummer is City Editor of the Daily Mail.