After the attack in the town of Macerata, the suspect got out of his car and made a fascist salute while wearing a tricolour scarf, according to Italian media.
He was allegedly found to be carrying a gun when he was arrested.
Six people including "people of colour" have allegedly been injured, four of them seriously, media reported.
One killed, five injured in grenade attack
Police said in a tweet that "the wounded persons are of foreign nationality".
The attack comes a day after a Nigerian asylum-seeker was arrested in Macerata over the gruesome killing of an 18-year-old Italian woman whose dismembered body was discovered in suitcases earlier this week.
However, no official link has been made between the two incidents.
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