First let me describe the norm in degree 0. Let $k$ be a commutative ring, given $\alpha\in H^0(BH,k)$, which corresponds to a an $H$ equivariant homotopy class of a map $EH_+\rightarrow k$ where $k$ has the trivial $H$ action. W can take the $n$th smash power of this map to obtain \begin{equation*} (EH_+)^{\wedge n}\cong E(H^n)_+\rightarrow k^{\wedge n}. \end{equation*} This map is naturally $\Sigma_n \wr H$ equivariant. Using the (equivariant) multiplication in $k$ we can make this map land in $k$ again. Now use the monomial representation above $G\rightarrow S_{G/H}\wr H\cong \Sigma_n \wr H$, we can regard this $\Sigma_n \wr H$ equivariant map as a $G$ map and $G$ acts freely on the source. Taking orbits we obtain an ordinary map $[BG_+, k]$ as desired.

To move up to higher degrees, I find it helpful to use equivariant spectra. So now our map $\alpha$ in degree $i$ will corresponde to a map $\Sigma^\infty_+ EH \rightarrow S^i \wedge Hk$. We proceed exactly as above but with one little snag. After taking smash powers the $S^i$ becomes $S^{i\rho}$ where $\rho$ is the standard $n$ dimensional representation of $\Sigma_n$ ($H$ acted trivially on $S^i$). When $i$ is even and $k$ is a field of characteristic $p$ (all $i$ in characteristic $2$) we use that $S^{i\rho}$ is $k$-orientable to land us in the correct degree.

For more details on the norm in spectra consult the article by May and Greenlees or Schwede's course notes especially chapter 7.

First let me describe the norm in degree 0. Let $k$ be a commutative ring, given $\alpha\in H^0(BH,k)$, which corresponds to a an $H$ equivariant homotopy class of a map $EH_+\rightarrow k$ where $k$ has the trivial $H$ action. W can take the $n$th smash power of this map to obtain \begin{equation*} (EH_+)^{\wedge n}\cong E(H^n)_+\rightarrow k^{\wedge n}. \end{equation*} This map is naturally $\Sigma_n \wr H$ equivariant. Using the (equivariant) multiplication in $k$ we can make this map land in $k$ again. Now use the monomial representation above $G\rightarrow S_{G/H}\wr H\cong \Sigma_n \wr H$, we can regard this $\Sigma_n \wr H$ equivariant map as a $G$ map and $G$ acts freely on the source. Taking orbits we obtain an ordinary map $[BG_+, k]$ as desired.

To move up to higher degrees, I find it helpful to use equivariant spectra. So now our map $\alpha$ in degree $i$ will corresponde to a map $\Sigma^\infty_+ EH \rightarrow S^i \wedge Hk$. We proceed exactly as above but with one little snag. After taking smash powers the $S^i$ becomes $S^{i\rho}$ where $\rho$ is the standard $n$ dimensional representation of $\Sigma_n$ ($H$ acted trivially on $S^i$). When $i$ is even and $k$ is a field of characteristic $p$ (all $i$ in characteristic $2$) we use that $S^{i\rho}$ is $k$-orientable to land us in the correct degree.

For more details on the norm in spectra consult the article by May and Greenlees or Schwede's course notes especially chapter 7.

First let me describe the norm in degree 0. Let $k$ be a commutative ring, given $\alpha\in H^0(BH,k)$, which corresponds to a an $H$ equivariant homotopy class of a map $EH_+\rightarrow k$ where $k$ has the trivial $H$ action. W can take the $n$th smash power of this map to obtain \begin{equation*} (EH_+)^{\wedge n}\cong E(H^n)_+\rightarrow k^{\wedge n}. \end{equation*} This map is naturally $\Sigma_n \wr H$ equivariant. Using the (equivariant) multiplication in $k$ we can make this map land in $k$ again. Now use the monomial representation above $G\rightarrow S_{G/H}\wr H\cong \Sigma_n \wr H$, we can regard this $\Sigma_n \wr H$ equivariant map as a $G$ map and $G$ acts freely on the source. Taking orbits we obtain an ordinary map $[BG_+, k]$ as desired.

To move up to higher degrees, I find it helpful to use equivariant spectra. So now our map $\alpha$ in degree $i$ will corresponde to a map $\Sigma^\infty_+ EH \rightarrow S^i \wedge Hk$. We proceed exactly as above but with one little snag. After taking smash powers the $S^i$ becomes $S^{i\rho}$ where $\rho$ is the standard $n$ dimensional representation of $\Sigma_n$ ($H$ acted trivially on $S^i$). When $i$ is even and $k$ is a field of characteristic $p$ (all $i$ in characteristic $2$) $S^{i\rho} \wedge Hk\simeq S^{in}\wedge Hk$ equivariantly which lands us in the correct degree. More generally we obtain $RO(G)$ graded coefficients.

For more details on the norm in spectra consult the article by May and Greenlees or Schwede's course notes especially chapter 7.

First let me describe the norm in degree 0. Let $k$ be a commutative ring, given $\alpha\in H^0(BH,k)$, which corresponds to a an $H$ equivariant homotopy class of a map $EH_+\rightarrow k$ where $k$ has the trivial $H$ action. W can take the $n$th smash power of this map to obtain \begin{equation*} (EH_+)^{\wedge n}\cong E(H^n)_+\rightarrow k^{\wedge n}. \end{equation*} This map is naturally $\Sigma_n \wr H$ equivariant. Using the (equivariant) multiplication in $k$ we can make this map land in $k$ again. Now use the monomial representation above $G\rightarrow S_{G/H}\wr H\cong \Sigma_n \wr H$, we can regard this $\Sigma_n \wr H$ equivariant map as a $G$ map and $G$ acts freely on the source. Taking orbits we obtain an ordinary map $[BG_+, k]$ as desired.

To move up to higher degrees, I find it helpful to use equivariant spectra. So now our map $\alpha$ in degree $i$ will corresponde to a map $\Sigma^\infty_+ EH \rightarrow S^i \wedge Hk$. We proceed exactly as above but with one little snag. After taking smash powers the $S^i$ becomes $S^{i\rho}$ where $\rho$ is the standard $n$ dimensional representation of $\Sigma_n$ ($H$ acted trivially on $S^i$). When $i$ is even and $k$ is a field of characteristic $p$ (all $i$ in characteristic $2$) we use that $S^{i\rho}$ is $k$-orientable to land us in the correct degree.

For more details on the norm in spectra consult the article by May and Greenlees or Schwede's course notes especially chapter 7.