On the wreath transfer in the homology of symmetric groups

For an odd prime $p$, we give an explicit formula for the wreath transfer $\mathrm{t}\mathrm{r}\text{\hspace{0.17em}⁣}:H_{\ast }({\mathrm{\Sigma }}_{p^2};{\mathbb{F}}_p)⟶H_{\ast }({\mathrm{\Sigma }}_p\wr {\mathrm{\Sigma }}_p;{\mathbb{F}}_p)$ associated with the standard inclusion ${\mathrm{\Sigma }}_p\wr {\mathrm{\Sigma }}_p\le {\mathrm{\Sigma }}_{p^2}$. 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.