Qatar is hardly the only gulf monarchy to allow open fund-raising by sheikhs that the United States government has linked to Al Qaeda’s Syrian franchise, the Nusra Front: Sheikh Ajmi and most of the others are based in Kuwait and readily tap donors in Saudi Arabia, sometimes even making their pitches on Saudi- and Kuwaiti-owned television networks. United States Treasury officials have singled out both Qatar and Kuwait as “permissive jurisdictions” for terrorist fund-raising.

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In many cases, several analysts said, Qatar has sought to balance a wager on the future of political Islam as a force in the region with a simultaneous desire not to alienate the West. It has turned a blind eye to private fund-raising for Qaeda-linked groups to buy weapons in Syria, for example, but it has not provided direct government funding or weapons. At times, Mr. Stephens and other analysts said, Western pressure has moved Qatar to at least partly suppress some of the overt fund-raising.

Qatar openly provides a base for leaders of the Palestinian militant group Hamas — deemed a terrorist organization by the United States and Israel — as well as money to help prop up its government in Gaza. But American and Israeli officials say Qatar has stopped short of providing the group with weapons, as Iran does.

Qatar has allowed members of the Taliban to open an office and make their homes in Doha, but as part of deals approved by Washington.

In Libya, Qatar and the United Arab Emirates are now backing rival sides in Libya’s escalating domestic unrest, each with unsavory ties: The U.A.E. is backing former fighters for Col. Muammar el-Qaddafi and members of his ruling elite, while Qatar is backing a coalition that includes militant Islamist groups.

During the 2011 uprising in Libya, Qatar supported an Islamist militia in Benghazi known as Rafallah al-Sehati that had relatively Western-friendly leaders but extremists in its ranks. The extremists later broke away to form Ansar al-Shariah, the militant group that played a role in the death of the American ambassador, J. Christopher Stevens.

Now Qatar is still backing militias at least loosely allied with the group in their fight against an anti-Islamist faction backed by the United Arab Emirates.