President Abd-Rabbu Mansour Hadi resigned on Thursday after Houthi fighters crushed his presidential guards and deployed outside his home.
Protesters marked the anniversary of the 2011 uprising that toppled Hosni Mubarak.
Qaradawi’s outspoken support for the Islamist movement has previously contributed to a diplomatic rift between Qatar and its Gulf Arab allies and Egypt.
Saudi Arabia’s Interior Minister, Prince Mohammed bin Nayef, was named Deputy Crown Prince on Friday.
Hadi said the group had a right to serve in posts in all state institutions, and a draft constitution that has been a source of disagreement was open to amendment.
Although the GCC described the Houthis’ actions as terrorism, they stopped short of promising any measure to counter the group in Sanaa.
Houthi leader Abdel-Malek al-Houthi warned President Abd-Rabbu Mansour Hadi that he had to implement a power-sharing deal.
AQAP claimed responsibility for the attack on Charlie Hebdo satirical newspaper in Paris this month that killed 17 people.
Rajab took a leading role in Shi’ite-led demonstrations in Bahrain in 2011 that demanded reforms.
Sheikh Ali Salman is facing four charges including promoting regime change by force, his defence lawyer said.
Qatar’s role as a host to exiled Brotherhood leaders has irritated Egypt, which considers the group a political threat.
The orders follow an attack on the Saudi border earlier this year leaving two guards and a senior military commander dead.
Ali al-Marri, who was convicted in 2009 of providing material support to al Qaeda, was released from federal prison in Colorado in the last few days.
The two sides have been talking about the visit for about 18 months, and a date had been tentatively set for last October.
Saudi Arabia is footing the budget of the King Abdullah bin Abdulaziz International Centre for Interreligious and Intercultural Dialogue in Austria.
The U.S. military has said it is planning to send more than 400 troops to train Syrian moderates outside the country.
Badawi was flogged 50 times last week but a second round of lashings was postponed for what a source told Reuters were medical reasons.
The discontent among Bahrain’s Shi’ite masses has worsened since the arrest of Sheikh Ali Salman, a Shi’ite Muslim cleric.
Walid Abu al-Khair, founder and director of watchdog group Monitor of Human Rights in Saudi Arabia (MHRSA), was sentenced last year to 10 years in jail.
Quoting from Muslim scholars, Sheikh Munajjid argued that to build a snowman was to create an image of a human being, an action considered sinful under Islam.
A witness said that protest marches turned violent, with dozens of young men throwing petrol bombs at security forces who fired bird shot and tear gas back.
Raif Badawi, who set up the “Free Saudi Liberals” website, was arrested in June 2012 and prosecuted for offences including cyber crime and disobeying his father.
The government confirmed last week that three of the four attackers, who carried out a gun attack on the country’s border, were Saudi nationals.
Terrorism experts said the hooded gunmen appeared to have carried out their attack methodically.
Saudi Arabia said in a statement via its official news agency that it denounces this “cowardly terrorist act”.
The blast appeared to target a group of students outside the college,
The witnesses said protest marches in Sheikh Salman’s district of the capital, turned violent, with security forces firing bird-shot and tear gas at the demonstrators.
King Abdullah said that lower prices were primarily caused by weakness in the global economy.
Sheikh Ali Salman, head of the al-Wefaq Islamic Society, was arrested after leading a protest rally against elections.
The royal court previously said that King Abudullah was suffering from pneumonia and temporarily needed help to breathe through a tube.