Members of the group shared photos on the Telegram network of the “martyr” posing in front of the Isis flag while holding a gun and wearing a suicide vest.
The attack earlier on Thursday saw two suicide bombers and gunmen attack crowds of Afghans as they flocked to Kabul’s airport.
Dozens were killed, including 11 Marines, a Navy medic and another military member, according to two US officials. They said another 12 service members were wounded and warned the toll could grow.
One of the bombers struck people standing knee-deep in a wastewater canal.
According to a statement posted by its Amaq propaganda agency in Arabic, the terror group claimed more than 150 people – including civilians, US forces and Taliban fighters – were killed or wounded in the suicide bombing.
Isis claimed the attacker came within five metres of American troops who were overseeing the collection of documents of those seeking to leave Afghanistan on evacuation flights before detonating the device.
A US official earlier said the attack was believed to have been carried out by Isis. The group’s affiliate in Afghanistan, known as Isis-K, is considered to be far more radical than the Taliban.
The name given to him by the group suggested he was an Afghan national from Logar province, south of the capital.
Isis’s last claim for an attack on the Taliban was on 14 August, the day before it took Kabul, according to The Independent’s security correspondent. There were six the previous month.
Western officials had warned of a major attack, urging people to leave the airport, but that advice went largely unheeded by Afghans desperate to escape the country in the last few days of an American-led evacuation before the US officially ends its 20-year presence on 31 August.
The US general overseeing the evacuation from Afghanistan has said the US will “go after” the perpetrators of the attacks if they can be found.
“We expect these attacks to continue,” said Gen Frank McKenzie, adding that Taliban commanders have been asked to take additional security measures to prevent another suicide bombing on the airport’s perimeter.
He said he had seen no indication that the Taliban allowed Thursday’s attacks to happen.
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