I and two of my colleagues are currently wrestling with Shapiro's lemma in the following situation. Let $G$ be a finite group, let $H$ be a subgroup of $G$, let $k$ be an algebraically closed field of characteristic zero, and $k^*$ its multiplicative group. We consider the standard normal complexes $C^\bullet{\left(G,\mathrm{Ind}^G_H(k^*)\right)}$ and $C^\bullet(H,k^*)$, where $$\mathrm{Ind}^G_H(k^*)=\left\{a:G\to k^*\Big|a(xh)=h^{-1}.(a(x))\ \forall x\in G\ \forall h\in H\right\},$$ with $G$-action on $\mathrm{Ind}^G_H(k^*)$ given by $(g.a)(x)=a{\left(g^{-1}x\right)}$ for $a\in \mathrm{Ind}^G_H(k^*),x,g\in G$. The specific version of the Shapiro map that we use is $\mathcal{S}_n:C^n{\left(G,\mathrm{Ind}^G_H(k^*)\right)}\to C^n(H,k^*)$ given by $${\left(\mathcal{S}_n(c)\right)}{\left(h_1,\dots,h_n\right)}=c{\left(h_1,\dots,h_n\right)}(e),\quad n\geq0.$$ We have been attempting to find an explicit formula for a quasi-inverse to this map. We also wish to determine whether or not this map exhibits a homotopy equivalence between the complexes in question. Any information on this would be appreciated.
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