Let $U$ be a normal subgroup of a group $G$ of finite index. On cohomology, somewhat dual to the functorially defined restriction map, $\text{res}^G_U\colon H^n(G, A) \to H^n(U, A)$, the finite index allows us to define a $0$-dimensional map that induces a correstriction $\text{cor}^G_H\colon H^n(U, A) \to H^n(G, A)$. This map has interesting properties relating it to $\text{res}^G_H$, such as the index formula and the double coset formula.

Has the same been done for the inflation map $\text{inf}^G_H \colon H^n(G/U, A^U) \to H^n(G, A)$? A way to define a $0$-dimensonal map that would induce a coinflation $\text{coinf}^G_H\colon H^n(G, A) \to H^n(G/U, A^U)$ with properties analogous to the correstriction?

Let $U$ be a normal subgroup of a group $G$ of finite index. On cohomology, somewhat dual to the functorially defined restriction map, $\text{res}^G_U\colon H^n(G, A) \to H^n(U, A)$, the finite index allows us to define a $0$-dimensional map that induces a correstriction $\text{cor}^G_H\colon H^n(U, A) \to H^n(G, A)$. This map has interesting properties relating it to $\text{res}^G_H$, such as the index formula and the double coset formula.

Has the same been done for the inflation map $\text{inf}^G_{G/U} \colon H^n(G/U, A^U) \to H^n(G, A)$? A way to define a $0$-dimensonal map that would induce a coinflation $$\text{coinf}^G_{G/U}\colon H^n(G, A) \to H^n(G/U, A^U)$$ on cohomology with properties analogous to the correstriction?