  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 $\mathbb
F_p$ vector space $V$, $G=\Sigma_{p^{n},p}$ a $p$-Sylow subgroup,
$\Sigma_{p^{n-1}}\int \Sigma_{p}$, or $\Sigma_{p}\int \Sigma_{p^{n-1}}$. Some
answers are given by natural invariants which are related to certain parabolic
subgroups. We also compute a free module basis for certain rings of invariants
over the classical Dickson algebra. This provides a computation of the image of
the 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 prove
that $\bar\tau^\ast=\xi$ in the ideal generated by the top Dickson algebra
generator.
