"I don't go in for golf much" means "not much at all," and you might even say "I don't go in for golf much; I generally don't like sports much at all."
So it ends up meaning that you don't like golf at all.
That means that it doesn't make sense to say you don't like it at all a lot.
You could possibly hear:
"Do you go in for golf much?"
"Well, I like it somewhat -- but I don't go in for it a lot." (or "not a whole lot")
"Not a lot" describes the degree to which you like something, so if you don't like it at all, there is no sense in describing exactly how MUCH you don't like it at all.
You can say:
"I go in for golf a lot."
"I go in for golf, but not a lot."
But I don't think you can say
"I don't go in for golf a lot."
Possibly you could say
"I don't really go in for golf a WHOLE lot" -- meaning "only a little, not a lot"
(Unless you are specifically negating or denying the proposition that you DO like it a lot, maybe.)
"Say! I hear that you go in for golf a lot!"
"No, not true. I don't go in for golf 'a lot.' I like it only a little."
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