Cool headed girl with quality (5)
I believe the answer is:
class
'quality' is the definition.
(class is a kind of quality)
'cool headed girl' is the wordplay.
'cool' becomes 'cl' (I can't explain this - if you can you should believe this answer much more).
'headed' means to remove the first letter (head can mean to remove the top from something).
'girl' becomes 'lass' (lass can informally mean a girl).
'lass' with its first letter taken away is 'ass'.
'cl'+'ass'='CLASS'
'with' is the link.
Can you help me to learn more?
(Other definitions for class that I've seen before include "Set, category" , "Order - quality" , "Category; form" , "Category; lesson" , "pupils?" .)