Fail to include old American institute (4)
I believe the answer is:
omit
'fail to include' is the definition.
(I've seen this before)
'old american institute' is the wordplay.
'old' becomes 'o' (common abbreviation eg in OE for Old English).
'american institute' becomes 'MIT' (Massachusetts Institute of Technology).
'o'+'mit'='OMIT'
(Other definitions for omit that I've seen before include "Miss out" , "Leave out or neglect" , "Fail (to do something)" , "Don't include" , "by-pass" .)