Wednesday's explosion killed Aushev's mother-in-law and brother-in-law, said Kaloi Akhilgov, a spokesman for the Ingush president.
Two other people in the car, Aushev's widow and another one of his brothers, were injured, Akhilgov told The Moscow Times on Thursday. He couldn't identify any of the victims.
The car was running on propane gas, which exploded when police fired at the car, Akhilgov said.
He denied reports on Interfax and RIA-Novosti, which cited local police as saying the explosion was caused by a bomb in the car and that the dead included Aushev's widow and adult son. The reports said the two injured people were Aushev's adult son and daughter.
Ingush police have said they opened fire after the car approached a checkpoint outside Nazran, made a U-turn, and sped away.
Aushev, 43, who campaigned against abductions by security forces in Ingushetia, died at the wheel of his car after unknown assailants peppered it with bullets on Oct. 25 in Kabardino-Balkaria.
U.S. Secretary of State Hillary Clinton recognized Aushev's work earlier this month by posthumously awarding him with the State Department’s annual Human Rights Defenders Award.
Suicide bombings at police checkpoints have increased in frequency in recent months in Ingushetia. A suicide car bomber struck a group of policemen at a checkpoint in Nazran on Thursday, killing himself and wounding at least 11 officers, The Associated Press reported.
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