Institution providing outlet after study (7)
I believe the answer is:
convent
'institution' is the definition.
(I know that convent is a type of institution)
'outlet after study' is the wordplay.
'outlet' becomes 'vent' (I've seen this before).
'after' says to put letters next to each other.
'study' becomes 'con' (con can informally mean to study).
'vent' after 'con' is 'CONVENT'.
'providing' is the link.
(Other definitions for convent that I've seen before include "Building for nuns" , "Religious house" , "Community of nuns" , "Building for prayer" , "Nunnery" .)