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?