Let $\pi\colon G'\to G$ be a surjective homomorphism of finite groups, and let $A$ be a $G$-module. I need explicit formulas for the inflation maps $${\rm Inf}^{r}\colon H^{r}(G,A)\to H^{r}(G',A)$$ for $r=0,-1,-2$. In this post, by $H^r(G,A)$ I mean the modified (Tate) cohomology in degree $r$.
I recall the definitions of $H^{-1}$, $H^0$, and $H^1$. Let $$A^G=\{a\in A\mid g\cdot a=a\ \,\forall g\in G\}$$ denote the subgroup of $G$-invariants, and $$A_G=A\,/\langle g\cdot a-a\mid g\in G,\, a\in A\rangle$$ denote the group $G$-coinvariants. Consider the map $$N\colon A\to A, \quad a\mapsto \sum_{g\in G} g\cdot a.$$ This map $N$ factors through a map $$N_*\colon A_G\to A^G.$$ We set $$ H^{-1}(G,A)=\ker N_*\,,\quad H^0(G,A)={\rm coker\,} N_*\,.$$ Moreover, $$H^1(G,A)=Z^1(G,A)/B^1(G,A),$$ where \begin{align*} &Z^1(G,A)=\big\{z\colon G\to A\,\mid\, z(g_1g_2)=z(g_1) +g_1\cdot z(g_2)\big\}\\ &B^1(G,A)=\{g\mapsto g\cdot a -a \,\mid\, a\in A\}\subseteq Z^1(G,A). \end{align*}
The inflation map on 1-cocycles is defined by $$ z\mapsto z\circ\pi\colon\, Z^1(G,A)\to Z^1(G',A).$$ This map induces the inflation map on $H^1$. The inflation maps on $H^r$ for $r<1$ are defined using dimension shifting. I understand that it is just an exercise to compute ${\rm Inf}^r$, but I struggle with this exercise!
Question 1. Is it true that the inflation map ${\rm Inf}^{-1}$ fits into the following commutative diagram: $\require{AMScd}$ \begin{CD} H^{-1}(G,A) @>{\rm Inf}^{-1}>> H^{-1}(G',A)\\ @VVV @VVV \\ A_G @= A_{G'} \end{CD}
Question 2. How can one explicitly describe the inflation map ${\rm Inf}^0$?
Question 3. How can one explicitly describe the group $H^{-2}(G,A)=H_1(G,A)$ and the inflation map ${\rm Inf}^{-2}$ in terms of (nonhomogeneous) cocycles?
