It followed a suicide bombing that killed more than 100 Afghans and 13 US soldiers as American troops evacuated in August.
AFP takes a look at the two groups and how their rivalry is likely to play out.
Who is IS-K?
The broader Islamic State group was officially founded in late 2014, when Sunni extremists fighting insurgencies in Iraq and Syria swore allegiance to a "caliphate".
Supposedly the heartland of a future universal Muslim homeland under the IS black banner, Abu Bakr al-Baghdadi's group seized a chunk of Iraq and eastern Syria.
This territory was eventually recaptured by US-backed forces, but not before it had inspired spin-offs elsewhere, including in "Khorasan", a region taking in parts of Afghanistan, Iran, Pakistan and Turkmenistan.
Jean-Luc Marret, of French think tank the Foundation for Strategic Research, describes IS-K as "a conglomeration of former jihadist organisations, including Uyghurs and Uzbeks, and Taliban defectors".
IS-K claimed Friday's bombing in Kunduz was carried out by a Uyghur, a member of China's persecuted Muslim minority, underlining the regional nature of the threat.
According to UN estimates, IS-K has between 500 and a few thousand fighters in northern and eastern Afghanistan, including cells under the nose of the Taliban in the capital Kabul.
Since 2020, the group has been reputedly led by one "Shahab al-Mujahir", whose nom de guerre suggests he arrived in the region from the Arab world but his origins remain murky.
He is variously rumoured to have been an Al-Qaeda commander or a former member of the Haqqani network, now one of the most powerful and feared factions in the Afghan Taliban.
How great is the threat
Up until 2020, overshadowed by the Taliban and targeted by a campaign of US air and drone strikes, the IS-K faction was losing influence.
But the arrival of the mysterious new leader seems to have marked a change in its fortunes.
According to researcher Abdul Sayed of online extremism tracker ExTrac, Shahad placed "a renewed emphasis on urban warfare and symbolic violence".
"Although the Taliban are its primary target, IS-K has chosen soft targets like religious places, educational institutions, and public places like hospitals, etc. to spread fears of its terrorism," Sayed said.
The Taliban and IS-K are both Sunni militant groups but, while the new Taliban-led regime in Kabul has promised to protect the minority Shiites, its rival remains bent on eradicating "apostates" and "hypocrites".
As in Iraq, where the original IS targeted Shiite communities to foment sectarian war, in Afghanistan IS-K has threatened the Hazara, a mainly Shiite ethnic minority.
Where did the rivalry begin?
Many of the fighters in IS-K have fought for the Taliban or allied groups, or come from insurgent movements inspired by Al-Qaeda. But now the groups' strategies have diverged.
The Taliban of 2021 has the goal of ruling Afghanistan under its interpretation of Islamic law, whereas IS-K is still wedded to the distant goal of a global "caliphate".
Taliban spokesmen brand the group "takfiri" -- Muslims who take it upon themselves to brand others apostates and thus condemn them to death -- while IS-K propaganda paints their rivals as sell-outs to the Americans.
But while the rhetoric is bloodcurdling, the border between the groups is porous, and fighters can shift sides as their commanders' views and opportunities evolve.
"IS-K has been previously successful in recruiting members disaffected with the Taliban and those who perceive the Taliban as too moderate," said Barbara Kelemen, of Dragonfly Security Intelligence.
"With the Taliban now seemingly implementing some moderate reforms to its rule, there is a high probability the group will try to capitalise on its position as the main rejectionist group in Afghanistan to recruit more disaffected former Taliban supporters and to mount attacks against the Taliban."
Do the Taliban have the upper hand?
"The Taliban's main message to the Afghan population since August 15 is that it has restored stability by ending the war," said Michael Kugelman of US think tank the Woodrow Wilson Center.
"But terrorist attacks like the one in Kunduz undermine that narrative in a big way," he warned.
Afghanistan's ousted US-backed government received hundreds of billions of dollars in support and security assistance and was backed by Western forces but could defeat neither the Taliban nor IS-K.
Now, the Taliban faces its rival with very little outside assistance, and none of the sophisticated intelligence gathering and surveillance equipment deployed by the US military.
They do know their enemy and the terrain though, and last week announced the destruction of an IS-K cell in Kabul in the aftermath of a suicide attack on the city's second-biggest mosque.
And they have the potential support of two groups that know IS-K's tactics very well.
As a report from the US-based Soufan Centre explained, "To combat IS-K, the Taliban is going to rely on the Haqqani network, Al-Qaeda, and other violent non-state actors for manpower, combat expertise, and logistical support."