ASCAP lists a song called THERE'S A FLEA IN LIZZIE'S EAR with the alternate title GO GET THE AXE. It was allegedly written by Robert Melvin Schultz, Jr., and published by Beam Me Up Music c/o Warner Brothers. However, if you look at a list of songs by Mr. Schultz, you will see quite a few familiar titles which I'm sure didn't originate with him. He's probably the kind of guy who claims a copyright when he makes a minor change to a traditional song.

Allmusic.com lists a recording of GO GET THE AXE by Randy Kaplan, on the album "Boyish Hips," 1997, where it is called "traditional." Here's an excerpt:

...get the axe; there's a flea in Lizzie's ear.
A boy's best friend is his mother.
Peepin' through the knothole of grandpa's wooden leg,
Why did they build the shore so near the sea?
Who cut the sleeves off of dear old Daddy's vest...?

Other albums containing GET THE AXE or GO GET THE AXE, according to Gracenote.com:
"Obscenity in the Milk" by Milhouse.
"Homeschooler's Folksong Collection" by The Homestead Pickers.
"150 Sing-Alongs" by Baby Basics.