Shapiro's lemma in the language of group extensions

Question

I am trying to understand Shapiro's lemma for $H^2$ in the concrete language of extensions of finite groups.

Let $H$ be a subgroup of a finite group $G$, and let $A$ be an $H$-module. Let ${\rm Ind}_G^H(A)$ be the induced module (see Serre, Galois Cohomology, Ch. I, 2.5). Shapiro's lemma says that the inclusion $H\hookrightarrow G$ and the map $\pi\colon {\rm Ind}_G^H(A)\to A$ of evaluation at $1$ give an isomorphism $H^2(G,{\rm Ind}_G^H(A))\simeq H^2(H,A)$ (of course, this is true in any degree).

Now let us view elements of $H^2$ as equivalence classes of extensions. An extension $0\to {\rm Ind}_G^H(A)\to \hat G\xrightarrow{f}G\to 1$ should correspond under this isomorphism to the extension $0\to A\to f^{-1}(H)/{\rm Ker}(\pi)\to H\to1$.

My question is how to realize the inverse map explicitely:
That is, starting from an extension $0\to A\to \hat H\to H\to1$, what is the extension of $G$ by ${\rm Ind}_G^H(A)$ corresponding to it under Shapiro's lemma?

Answer 1

Shapiro's Lemma boils down to the following isomorphism for a subgroup of finite index: Let us write $G=\bigcup_{i=1}^t g_iH$ for left coset representatives of $H$ in $G$. For a $G$-module $P$ we have the following isomorphism $$\Psi:Hom_H(P,A)\cong Hom_G(P,Ind_H^G(A))$$ $$ \Psi(f)(p) = \sum_{i}g_i\otimes f(g_i^{-1}p).$$ If we think now of the extension $$1\to A\to \hat{H}\to H\to 1$$ as given by a two cocycle $\beta:H\times H\to A$, we can proceed in the following way: We have the Bar Resolutions $B_H$ and $B_G$ of the trivial module $\mathbb{Z}$ over $H$ and $G$ respectively. The two cocycle $\beta$ can then be considered as an element in $Hom_H((B_H)_2,A)$. Since $B_G$ can also be considered as a resolution for $\mathbb{Z}$ over $H$, we will have a lifting of the identity map $B_G\to B_H$ as a map of $H$-modules. This will already gives us a two cocycle in $Hom_H((B_G)_2,A)$. Now use the above isomorphism to get a two cocycle in $Hom_G((B_G)_2,Ind_H^G(A))$. This will give you the desired cocycle and therefore the desired extension.

The main computational difficulty here (besides calculating the isomorphism $\Psi$) is in lifting the identity map $\mathbb{Z}\to\mathbb{Z}$ to $B_G\to B_H$. In degree zero this is rather simple: you choose right coset representatives $g_i'$ and then map $hg_i'[]\in (B_G)_0$ to $h[]\in (B_H)_0$. In degrees one and two it will be more complicated. In many concrete cases you can calculate explicitly projective resolutions for $H$ and for $G$. Also, in case $H$ is normal in $G$ you can write things more concretely, by choosing a two cocycle for $G/H$ with values in $H$ representing the extension $$1\to H\to G\to G/H\to 1$$ (even if $H$ is not abelian).

In the general case I do not think that there is a neater way of writing this down. Of course, I will be more than happy to be proven wrong about this.