In the theory of cohomology of groups there are several important maps between cohomologies: restriction, inflation, transfer and Evens norm. The first three of them can be clearly seen on the level of the derived categories and derived functors:

1) Let $H$ be a subgroup of $G.$ Then we have the restriction functor $\downarrow^G_H\:: kG$-${\rm Mod}\to kH$-${\rm Mod}$ which is exact, so we can extend it on the derived category $\downarrow^G_H\:: \mathcal{D}(kG)\to \mathcal{D}(kH)$. Since $Ext^n_{?}(M,N)=Hom_{\mathcal{D}(?)}(M,N[n])$ this functor induces a map $$res:Ext^n_{kG}(M,N)\to Ext^n_{kH}(M,N)$$

2) Let $H$ be a normal subgroup of $G.$ Then similarly we have the restriction functor $\downarrow_G: \mathcal{D}(k[G/H])\to \mathcal{D}(kG)$ and it induces a map $inf: Ext^n_{k[G/H]}(M,N)\to Ext^n_{kG}(M,N).$

3) Let $H$ be a subgroup of $G$ and $[G:H]<\infty.$ Then we have functors $Hom_{kG}$ and $Hom_{kH}$ between categories $ kG\text{-}{\rm Mod}^{\rm op} \times kG\text{-}{\rm Mod} \to k\text{-}{\rm Mod}$ and a natural transformation ${\rm Tr}: Hom_{kH}\to Hom_{kG}$ by the formula ${\rm Tr}(\varphi)=\sum\limits_{gH\in G/H} g\varphi.$ It yields a natural transformation between derived functors $${\rm Tr}:{\bf R}Hom_{kH}(-\downarrow,=\downarrow)\to {\bf R}Hom_{kG}(-,=):\mathcal{D}(kG)^{\rm op}\times \mathcal{D}(kG)\to \mathcal{D}(k).$$ So we get a natural map ${\rm Tr}:Ext^n_{kH}\to Ext^n_{kG}.$

So my question is: What about Evens norm? How to see it on the level of derived categories?

It looks like one has to consider "derived functor" of the tensor induction functor, but it is not even an additive functor, it is a polynomial functor of degree $[G:H]$. As far as I know, the theory of model categories enables us to consider derived functors of such functors, but I don't know this theory well enough.