The well-known embedding theorem for countable groups by Higman-Neumann-Neumann says that
Theorem 1: Every countable group can be embedded in a 2-generated group.
One can prove this theorem using wreath product (instead of HNN-extensions) in order to preserve solvability, see Embedding Theorems for groups, Neumann-Neumann 1959
I know that with a similar but not identical construction one can obtain an analogous result for amenable countable groups:
Theorem 2: Every amenable countable group embeds in a 2-generated amenable group.
See HNN Embedding Theorem for Amenable Groups?
My question is the following: Is there a countable amenable group $G$ such that the Neumann-Neumann construction (theorem 1) fails to preserve amenability?