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In this work we compute the induced transfer map: $$\bar\tau^\ast:\func{Im}(res^\ast:H^\ast(G) \to H^\ast(V)) \to \func{Im}(res^\ast: H^\ast(\Sigma_{p^n}) \to H^\ast(V))$$ in $\func{mod}p$-cohomology. Here$\Sigma_{p^{n}}$ is the symmetric group acting on an $n$-dimensional $\mathbbF_p$ vector space $V$, $G=\Sigma_{p^{n},p}$ a $p$-Sylow subgroup,$\Sigma_{p^{n-1}}\int \Sigma_{p}$, or $\Sigma_{p}\int \Sigma_{p^{n-1}}$. Someanswers are given by natural invariants which are related to certain parabolicsubgroups. We also compute a free module basis for certain rings of invariantsover the classical Dickson algebra. This provides a computation of the image ofthe appropriate restriction map. Finally, if $ \xi:\func{Im}(res^\ast:H^\ast(G) \to H^\ast(V)) \to \func{Im}(res^\ast}:H^\ast(\Sigma_{p^n}) \to H^\ast(V)) $ is the natural epimorphism, then we provethat $\bar\tau^\ast=\xi$ in the ideal generated by the top Dickson algebragenerator.

The transfer in mod-p group cohomology between \Sigma_p \int \Sigma_{p^{n-1}}, \Sigma_{p^{n-1}} \int \Sigma_p and \Sigma_{p^n}