Just a quick clarification for non-brits about the term Secretary of State as used in the UK vs the US
The US Secretary of State is the single person who runs the State Department, which deals with foreign affairs. In the UK, any minister who runs a department (apart from the Treasury, which is run by the Chancellor) is a Secretary of State.
So Theresa May is Secretary of State for the Home Department (known as the Home Office and roughly equivalent to parts of the US Justice Department and Homeland Security). She is not "UK Secretary of State" in the US sense. The Home Office is responsible for the police, the Security Service (MI5), and the Border Force (although not the courts, which are administered by the Ministry of Justice). The Security Service (MI5) is roughly analogous to the FBI and Homeland Security's counter terrorism and counter espionage roles.
William Hague is Secretary of State for Foreign and Commonwealth Affairs, who runs the Foreign and Commonwealth Office, equivalent to John Kerry and the US State Department. The Foreign Office runs the Secret Intelligence Service (MI6) and GCHQ as these are both (theoretically) foreign intelligence services equivalent to the CIA and NSA.
This push by the government has not mentioned GCHQ, but has been all about giving the police more powers to request communications data. They have clearly decided that GCHQ has become a toxic brand for them.