Held on behalf
Section 3(2)(b) of FOIA states "For the purposes of this Act, information is held by a public authority if ... it is held by another person on behalf of the authority" 
There is a relevant ICO guidance note which gives the example of research material being held by another.
This ICO decision notice (FS50118044, Leeds City Council) is particularly relevant, where market research response data held by a private company was deemed to be held by the 3rd party on behalf of the public authority, and the Information Commissioner partly upheld the Section 3(2)(b) appeal.
There have also been a few cases at the Information Tribunal where Section 3(2)(b) was considered: