The country's government is at odds with the EU's refugee policies.
Turkish president said that EU has paid only two million of the promised three billion euros ($3.3 billion).
The country continues to reject Eritrean and Sudanese applications in the country.
Amaq's claim comes just days after the radical Islamist group said it inspired the Nice truck attack.
The rescued migrants from Nigeria, Ethiopia, Eritrea and Bangladesh were taken to a Sicilian port for questioning.
Uber Mexico said in a statement that it bore no responsibility but was cooperating with the authorities.
If the 65.3 million people were a single population, it would be the world's 21st largest.
The charity's report reveals the trauma children have suffered getting to Europe.
Two attackers were to detonate suicide vests while others were to kill passersby with guns, prosecutors say.
The large fishing boat rocks violently before capsizing, sending migrants tumbling into the sea.
Ankara is unable to fulfil all the EU's conditions for unrestricted travel.
Immigration Minster James Brokenshire proposes new Immigration Bill measures that will allow children to be dispersed across local authorities.
The charity says the EU cannot delegate responsibility to Turkey.
The U.N. found that some of the deported refugees were denied the chance to claim asylum.