Lashkar-e-Kahar (Army of the Imperious) said in a telephone call to Current News Service (CNS), a Kashmiri news agency, that it carried out the bombings in Varanasi.
"We have carried out the attacks," said a spokesman for the group who identified himself as Abdul Jabbar alias Abu Kahar on Thursday.
He threatened more attacks if "India does not stop atrocities against Kashmiri Muslims. Until that happens we will not allow people of India to live in peace".
Police said they had never heard of the group before but were taking the call seriously and had started investigations.
The authorities have said they suspected that "terrorists", a euphemism for Muslim fighters, were behind the carnage.
Counter-terrorism forces shot dead a wanted senior operative with the Muslim group Lashkar-e-Taiba (Army of the Pious) on Wednesday and were investigating his links to the Varanasi blasts.
Muslim-majority Kashmir is in the grip of an insurgency against Indian rule in which thousands of people have died in 16 years of violence.