The father of a Jordanian fighter pilot who was burned alive by the Islamic State of Iraq and the Levant (ISIL) group has demanded from Jordan a harsh and swift revenge for his son's murder.

Speaking to Al Jazeera on Wednesday, Safi al-Kassasbeh said that the execution of two al-Qaeda-linked prisoners that followed his son's death was not sufficient.

Jordan's King Abdullah II has returned to the capital Amman from the US to join the mourning in his country. He has vowed to go after the ISIL fighters.

Would-be Iraqi female suicide bomber Sajida al-Rishawi and Iraqi al-Qaeda member Ziad al-Karbouli were hanged at dawn on Wednesday, government spokesman Mohammad al-Momani said, after Jordan vowed an "earth-shattering" response to avenge the burning alive of pilot Moaz al-Kassasbeh.

"I demand none of them amongst us be kept alive. I demand the revenge be greater than executing prisoners. I demand the ISIL organisation be annihilated," Safi al-Kassasbeh told Al Jazeera in Karak, central Jordan.

"This murderous organisation, made up of militants from all the world countries, is acting in barbaric ways, violating all the international laws, codes of ethics, and prisoners' conventions. That is why I strongly demand the government to swiftly take revenge for the blood of Moaz and the dignity of our country."

Double execution

Jordan had promised to begin executing the prisoners on death row at daybreak in response to the murder of Kassasbeh, who was captured by ISIL when his plane went down in Syria in December.

Rishawi, 44, was condemned to death for her participation in deadly attacks in Amman in 2005 and ISIL had offered to spare Kassasbeh's life and free a Japanese hostage - who was later beheaded - if she were released.

Al Jazeera's Nisreen El-Shamayleh, reporting from Amman, said that the executions took place at 5am local time (3:00 GMT).

"Usually, it is a long and highly bureaucratic process to carry out executions in Jordan. Several ministries and the king should approve them," she said.

"However, a security source told Al Jazeera last week that Jordan would speed up the process if the pilot was harmed."

Karbouli was sentenced to death in 2007 on terrorism charges, including the killing of a Jordanian in Iraq.

Jordan had on Tuesday vowed to avenge the killing of Kassasbeh, hours after the harrowing video emerged online showing the caged 26-year-old F-16 fighter pilot engulfed in flames.

The video - the most brutal yet in a series of gruesome recorded killings of hostages by ISIL - prompted global revulsion and vows of continued international efforts to combat the Sunni group.

Jordan, a crucial ally of Washington in the Middle East, is one of five Arab countries that has joined a US-led coalition of countries carrying out air strikes against ISIL in Syria and Iraq.

'Vile murder'

Al-Azhar, Sunni Islam's most prestigious centre of learning, has called for the killing and crucifixion of ISIL fighters, expressing outrage over their murder of the pilot.

In a statement after the burning alive of Kassasbeh, the Cairo-based centre called for the "killing, crucifixion and chopping off the limbs of Islamic State terrorists".

Al Jazeera's Rula Amin, reporting from Amman, said that the killing of Kassasbeh has unified public opinion in Jordan, even from among those who opposed the country's participation in the US-led coalition against ISIL.

"The big line now is to strike, and to do whatever it takes to retaliate," our correspondent said.

Meanwhile, the European Union combined a statement of solidarity with Jordan over the killing, with criticism of its immediate execution of Rishawi and Karbouli.

"While all efforts must be made to counter terrorism and hold the perpetrators accountable, our reaction to the threat posed by (ISIL) needs to be consistent with our common values on justice and the rights of prisoners," foreign policy chief Federica Mogherini said in a statement on Wednesday.