The Islamist group, Boko Haram, says it carried out Thursday's bombing of Nigeria's police headquarters which left at least six people dead.
The attack comes days after Nigeria's police chief visited the north-eastern town of Maiduguri, where Boko Haram is based and vowed to defeat the group.
The police say one of those who died was a suicide bomber - if confirmed it would be Nigeria's first such attack.
Boko Haram wants to overthrow the state and implement Islamic law.
In 2009, hundreds of Boko Haram supporters, including the group's leader Mohammed Yusuf, were killed after they attacked police stations in Maiduguri and other northern towns.
The group has since reformed and in the past year has killed dozens of police officers, politicians and anyone who criticises it, including Christian preachers and clerics from other Muslim groups.
It has also said it planted several bombs which went off in Abuja and other states after President Goodluck Jonathan's inauguration last month.
"We are responsible for the bomb attack on the police headquarters in Abuja which was to prove a point to all those who doubt our capability," said a statement from the group.
President Goodluck Jonathan on Friday visited the scene of the blast and urged Nigerians not to panic.
He said all countries were affected by "terrorist attacks".
The president was accompanied by police chief Hafiz Ringim, who may have been the target of the attack.
The BBC's Jonah Fisher in Lagos says that, for the Nigerian authorities, the attack is an embarrassing strike at the very heart of their security establishment.
Inspector General Hafiz Ringim went to Maiduguri earlier this week, taking reinforcements and promising that the Boko Haram problem would be solved within months.
Our correspondent says the group's response was delivered directly to police head office in Abuja.
Less than two minutes after Inspector General Ringim arrived for work on Thursday morning, a car that had been following closely behind his vehicle exploded in the car park.
Our correspondent says it is still unclear whether the driver of the car intended to die in the blast.
The blast in the car park of the police base also destroyed many vehicles and a large plume of smoke could be seen rising from the scene.
Residents say the explosion was heard across the city.
"My windows were shaking and I heard the loud noise. I saw smoke coming up," Reuters news agency quotes one witness as telling a local television station.
A bus commuter who saw the blast told the BBC's Focus on Africa programme: "All of a sudden there was this loud explosion. Everybody was scared and people began to run around."
He said police and Red Cross officials had rushed to the scene, and had moved people away from the area.
The police said 33 cars had been damaged beyond repair and 40 more had been partially damaged by the explosion.
The Boko Haram sect accuses Nigeria's government of being corrupted by Western ideas and wants to overthrow the state and impose Islamic law on the country.
It has killed dozens of people, mostly shot by gunmen riding motorbikes, in Maiduguri, the capital of Borno state.