On the wreath transfer in the homology of symmetric groups

For an odd prime $p$, we give an explicit formula for the wreath transfer $\mathrm{tr} \colon H_*(\Sigma_{p^2};\mathbb{F}_p)\longrightarrow H_*(\Sigma_p\wr\Sigma_p;\mathbb{F}_p)$ associated with the standard inclusion $\Sigma_p\wr\Sigma_p\le\Sigma_{p^2}$.

The proof is a direct algebraic computation, using Cartan's classical double-coset formula together with a generating-function argument.

As an immediate consequence, we obtain an elementary proof of the validity of the mixed Adem relations in Goodwillie calculus.