Who decides which, if any, solutions are publicized to students? For students, some textbooks publicize no solutions, some just half (either odd or even numbered exercises), and some all. How's this decided? Does the author have the final say? Can publishers override an author? A debate between both?
I'm not asking about instructors' solution manual that divulges solutions just to instructors.
I would be surprised if any publisher required a particular outcome or, especially, overrode an author's decision. It would make for an awkward relationship with the author.
They might, however, make suggestions, since the commercial value of a book might be affected by the decision. But the commercial value and the educational value might be at odds. I've never had a discussion with an editor about this subject and have published with several.
I do know of one case where a grad student was paid to produce solutions. I believe the funding came from the publisher, but it has been too long ago to remember details. But this is something an author could request of the editor. However, such payments usually come out of future royalties to the author, I'd expect.