A perfect-base alternating wreath product is generated by elements of order three

For any perfect group $H$ and $m\ge 3$, the permutation wreath product $H\mathrm{wr}{A}_m$ is generated by elements of order three. The top $A_m$ is generated by 3-cycles. With $\sigma =(123)$, the product of the conjugate $h_1\sigma h_1^{-1}$ and ${\sigma }^{-1}$ is $h_1{h}_2^{-1}$; using ${\sigma }^{-1}$ gives $h_1{h}_3^{-1}$. The commutator of $h_1{h}_2^{-1}$ and $k_1{k}_3^{-1}$ is $[h,k{]}_1$, since distinct coordinate groups commute. Perfectness gives the entire first coordinate group, and transitivity gives all coordinates. No finite-generation assumption on $H$ is needed.