Who produced The Who's second album?

Merely idle curiosity. What brought it to mind was a bit of musing on the differences between their first album, My Generation, and their second, A Quick One. (which I think might have originally been titled "Happy Jack". Not that that really matters.)

To start with, their first album was recorded (or over-recorded) rather poorly. It's noisy. I think I read once that that album was recorded in a hurry. Not a big surprise, they had yet to prove themselves an act that could sell a lot of discs. The engineers were probably underpaid rookies. If I remember right the producer was a guy named Shel Talmy. I don't know anything else about him except that I don't think he lasted long. That album also contained songs that sounded much more like American R&B than British Invasion. You can imagine Mitch Ryder and The Detroit Wheels doing some of those songs.

Their second album has much cleaner sound. It also sounds a lot more "Mod". Glyn Johns was their producer all through their heyday. Is this when he joined up?

While we're on the subject of producers, what exactly is their role? I've heard that part of their role is to come up with canned sounds that the artist might need. But there must be more to it than that.

I suspect that he's also the head sound engineer. Maybe the only engineer is some cases. I've seen hints before that the producer also has authority over the artist, at least in those cases of artists who have not proven themselves money-makers yet. (Though there must be some give and take there. It's hard to picture George Martin actually being boss over The Beatles. But maybe he could have vetoed a song that he really objected to?)

Perhaps the producer is more the agent of the record label than of the artist? Is he, maybe, the one with the responsibility to provide the lable with a marketable product? Something that is recorded properly and that won't get them in trouble over copyright issues or with legislators, Mothers Against Obscene Songs and stuff like that? Would that be the producers function?