The home secretary Theresa May has gone public with direct criticisms of education secretary Michael Gove's handling of the Trojan Horse affair – suggesting an internal cabinet tussle over who can be toughest on threats of extremism.

In a letter released on Tuesday night, May upbraided Gove for the Department for Education's handling of allegations regarding Islamists in Birmingham schools for what she called "the inability of local and central government to tackle the problem effectively".

In the letter May asks Gove: "Is it true that Birmingham city council was warned about these allegations in 2008? Is it true that the Department for Education was warned in 2010? If so, why did nobody act?"

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