Wow! I totally remember that!
Yes, I remember I used to love this sketch as a child! And yes, I do remember seeing quite a bit in the late 1980s. My brothers and I used to have this song pretty well memorized.
Henry: "There's a hole in the bucket, Dear Liza, Dear Liza,
There's a hole in the bucket, Dear Liza, a hole."
Liza: "Well, fix it, dear Henry, dear Henry, dear Henry,
Well, fix it, dear Henry, dear Henry, well fix it!"
Henry: "With what shall I fix it? Dear Liza, Dear Liza
With what shall I fix it? Dear Liza, with what? (hmm.. huh?)
Liza: "With a stone, dear Henry, dear Henry, dear Henry
With a stone, dear Henry, dear Henry, with a stone"
This goes on for quite a while until with a hilarious ending. Henry sings kind of slowly and Liza sings really fast. The lyrics probably are available somewhere, but it would be cool to find either an audio or video recording of this. It's very cool and quite funny.