A British Islamic State fighter has called for the death of Home Secretary Theresa May following a speech she made yesterday in which she called on British Muslims to help defeat extremism.

The jihadist Abu Abdullah Britani – who is believed to be Abu Rahin Aziz, a former credit control operator from Luton – made the threats on Twitter, calling for Muslims to "hunt down" and "kill" Mrs May. Among other abusive tweets he wrote "#TheresaMay responsible for many Muslims behind bars, we should track this cow & tie to back of car & dragged through London" and "#TheresaMay a day will come when someone crosses your path & terror will be struck in your heart."

Announcing the counter-terrorism strategy which a Conservative government would implement, Mrs May said yesterday: "The foundation stone of our new strategy is the proud promotion of British values."

She stressed that she did not expect everyone to watch Coronation Street or for a "flag to be flown from every building or demanding that everyone drinks Yorkshire Tea". However, she continued: "These values – such as regard for the rule of law, participation in and acceptance of democracy, equality, free speech and respect for minorities – are supported by the overwhelming majority of British people.

"They are sustained by our most important local and national institutions.

"And they are the means by which we have made our multi-racial, multi-cultural and multi-religious society succeed.

"They are the values that unite us. They are what allow us all to enjoy our individual freedoms, to lead varied lives, to live in diverse communities and to do all of these things while still living peacefully, respectfully and harmoniously alongside one another."

She urged the involvement of the whole community, saying: "Government cannot act alone. Individual people, families and whole communities need help and those of you fighting the extremists deserve our support. So my invitation is clear – come and join that partnership. If you join us, we will do everything we can to help you.

"We will help you to stand up to the extremists by denying them the opportunity to spread their messages of hate and division."

In a warning to those who rejected British values, she said: "The game is up. We will no longer tolerate your behaviour. We will expose your hateful beliefs for what they are."

She stressed: "We have been clear all along that the government's counter-extremism strategy must seek to defeat extremism in all its forms, but it's obvious from the evidence that the most serious and widespread form of extremism we need to confront is Islamist extremism."

Among the concrete measures she put forward were an independent figure to investigate the application of Sharia law in England and Wales, an investigation of the police response to honour crimes, female genital mutilation and forced marriage, and a requirement that all foreign religious workers in pastoral care roles would speak English. There would be a sharp reduction in funding for translations services and a boost to English language training. There would also be banning orders for extremist groups and 'extremism disruption orders', civil powers to be used against extremists.

May was recently forced to drop rules under which ministers could order universities to ban extremist speakers, under pressure from Liberal Democrat colleagues worried about the implications for freedom of speech.

Worries about the extent to which young people in particular are being radicalised by Islamist teaching have been fuelled by news today that nine British medical workers have crossed into Syria to work with Islamic State. They had all been studying in Sudan.

Responding to the Home Secretary's speech, the Muslim Council of Britain said: "We have consistently argued that the best counter-terrorism strategy involves upholding our own values of freedom, civil liberties and ensuring that we do not single out any specific community. When this government came to power, there was a sense of optimism within communities that it would uphold and champion these values.

"However, whilst Mrs May rightly speaks for values promoting the rule of law, participation in and acceptance of democracy, equality, free speech and respect for minorities, it is disappointing to see that her proposals seem to be at variance with the very qualities she wants us to aspire to."

The Home Secretary's proposals, it said, "are designed more to address the anxieties people have against Muslims and their religious life, rather than the scourge of terrorism itself". It warned against dividing the Muslim community and said that most extremist recruitment took place outside mosques and online.