Crazy nocturnal flyers (4)

I believe the answer is:

bats

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This could be a double definition.

'crazy' is the first definition.
(bats can informally mean crazy)

'nocturnal flyers' is the second definition.
'bats' can be an answer for 'flyers' (things that fly). I am not sure about the 'nocturnal' bit.

Can you help me to learn more?

(Other definitions for bats that I've seen before include "Clubs; flying mammals" , "Loopy; mammals" , "Bonkers (sl.)" , "Cricketers" , "out to lunch" .)

I've seen this clue in The Times.

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