Aksyonov told Russian state television the death toll from the attack now stood at 18, up from a previous estimate of 13. The shooter, who committed suicide afterward, apparently was not included in that number.
The town of Kerch is on the western end of a new bridge between Russian Federation and Crimea.
The Telegram Channel Mash, which is linked to the Russian police, published a photo of a young blond man, armed with a rifle - supposedly a screenshot from security cameras.
The Investigative Committee identified the attacker as Vladislav Roslyakov, 18. The committee said all the victims have died of gunshot wounds, contrasting with previous statements by other officials saying they had wounds resulting from an explosion.
The explosion went off at about midday (0900 UTC/GMT).
After Wednesday's attack, local officials declared a state of emergency on the Black Sea peninsula and cordons of Russia's National Guard circled the school. But witnesses reported that at least some victims were killed in an attack by a gunman or gunmen. "The motives and versions of this tragedy are being thoroughly studied", Putin said following the talks with the Egyptian president.
He promised that the government will do everything necessary to help those wounded.
Pupils and college staff were among those killed, she said. The college's website says it accepts students in the last three grades of school, from the age of 14.
Sergei Melikov said the explosion had been caused by an improvised explosive device.
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Another witness who gave his name as Sergei and who worked nearby said in a video on the Kerch.tv website that he heard "a bang and shots".
Officials said they are treating the situation now as a mass murder and not a terrorist attack.
Russian media reported that an unidentified gunman or gunmen attacked the college.
But Roslyakov confirmed earlier reports that the teenager had acquired a gun license legally. and promised a transparent investigation with "no secrets".
The college is for students who have left high school and are studying for a technical trade.
Explosives experts were inspecting the college building for other possible bombs, according to Anti-Terrorism Committee spokesman Andrei Przhezdomsky.
The blast was caused by a bomb packed with metal objects and was set off in the cafeteria, investigators said.
Anastasia Yenshina, a 15-year-old student at the college, said she was in the bathroom when the ground shook from an explosion.