A reactionary government will liberalise the economy and abolish democracy. It must formalise power, so that the people who appear to hold power are the same as the people who actually do.
How will it cement its power?
It must write a new national curriculum to explain the principles of non-democracy. History lessons will demonstrate the unusualness of democracy. Novels such as "Starship Troopers" could be studied in English Literature lessons.
The state broadcaster will encourage self-reliance instead of socialism. Universities will produce research designed to support the self-reliance and the new state over socialism. Control of university funding will encourage such research.
If the state broadcaster and the universities cannot be reformed, they will have to be dissolved, like the monasteries, as competing power centres.
The state schools will remain, because they definitely can be reformed.
The armed forces will also have to be reeducated to ensure loyalty to the new principles of the state. This should not be difficult. Different ethics classes, and a different selection of military history, will show that taking the necessary measures to preserve the government from the people are both moral and easy.