A source close to the Education Secretary said: "Michael Gove thinks Theresa May is a superb Home Secretary.
"We will continue to work well with the Home Office and other government departments to combat extremism in all its forms.
Ofsted will publish their findings next week and Peter Clarke will publish his report in July."
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He is accused of raudulently attempting to gain money and housing by pretending he had lost family members.
A plan will be outlined on Monday by Northern Ireland Secretary James Brokenshire after the fourth deadline is missed.
The Government intends to fund abortions in England for women arriving from Northern Ireland, where it is illegal, Philip Hammond has said.