The two inquiries, announced by Home Secretary Theresa May, will investigate alleged child abuse by MPs and wider public institutions.
The leaking, from her department, of a letter to Michael Gove, raises serious questions for the Home Secretary to answer.
One school was praised by Ofsted for using money from the pupil premium to pay for a number of cultural trips to Saudi Arabia.
ITV News has seen an Ofsted report that strongly criticises the governance of one of the Academy schools at the centre of the 'Trojan Horse' investigation into possible links with hard-line Islamists.
The says measures to keep students safe are "inadequate", while the school's equalities policy was deemed "not fit for purpose".
The regulator also warns that "leaders and governors are not doing enough to mitigate against cultural isolation", while female staff complained about feeling "intimidated" by male colleagues,
Some of the key points raised in the report:
These concerns, along with several problems with the teaching and curriculum, have led to the school being placed in 'special measures' by the regulator,
This means the school has to follow an action plan to improve its performance or risks being closed down.
Education Secretary Michael Gove has denied being at war with Theresa May, saying the Home Secretary is doing a "fantastic job".
Asked if she was too soft on Islamic fundamentalism, Mr Gove told reporters: "No absolutely not, she's doing a very fine job."
The Prime Minister has stepped in after an exchange between ministers over the handling of alleged Islamist extremism in Birmingham schools.
A Downing Street source said David Cameron was "keen to establish the facts" after Theresa May wrote to Michael Gove questioning why his department had not acted on warnings of a supposed plot by Islamist extremists.
May's letter inquired about evidence that the Department for Education knew of the 'Trojan Horse' allegations in 2010 and asked: "If so, why did nobody act?"
As the letter emerged the pair took the unusual step of issuing a joint statement insisting they were "working together" on the issue.
From the Government's point of view, too much of today's news agenda has been taken up discussing the spat between Theresa May and Michael Gove.
Theresa May is much-beloved on the Tory right, seen as a big success as Home Secretary and is leading the current polls for potential future leader of her party.
Michael Gove is very close to David Cameron and is leading - in Tory terms - critical education reforms.
They're both effectively unsackable members of the Cabinet.
If that wasn't the case, I think David Cameron would be thinking about sacking them for this extraordinary case of ill-discipline at a terrible time for the Government.
Extremism "anywhere in society is a serious problem" that is being tackled by ministers across departments, a Government spokesman said.
The message was sent out after an exchange between Theresa May and Michael Gove over the handling of the threat of Islamist extremism in schools in Birmingham.
There is no difference between the Education Secretary and the Home Secretary, who are both working energetically together to tackle the challenge posed by any form of extremism.
The spokesman added that the work of the Prime Minister's Extremism Taskforce demonstrated action being taken by government.
Speaking to BBC Radio 4's World At One, Conservative Party chairman Grant Shapps denied that the letter from May to Gove demonstrated a fight within the cabinet. He said:
It's somehow being suggested by commentators that this is somehow a big bust-up. In fact, I see Michael and Theresa work together all the time.
A spokesperson for Michael Gove and Theresa May said:
The Department for Education and the Home Office take the problems in Birmingham schools and all issues relating to extremism very seriously.
Michael Gove and Theresa May are working together to ensure we get to the bottom of what has happened in Birmingham and take the necessary steps to fix it.
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."