Sit an exam vs sit for an exam

Hi everybody,

Could you tell me whether these two word-groups differ or not (sit an exam and sit for an exam)? Based on dictionary definitions, these are the same, i.e. 'take an exam' (both UK English).

'go in for' also seems the same (and also seems UK). Does it differ from the previous ones?

How can I indicate that I would like to participate in an exam? 'to register for an exam'? (even if it is good, I'm curious about other word-groups which can express the same idea).

Please, help me understand these and fill the gaps in my vocabulary.

Thanks for your answer in advance.