inflation map
Inflation is the natural way to lift cohomology from a quotient group up to the whole group. If you have a normal subgroup N and a cocycle defined on the quotient G/N, you can pull it back to the whole group by simply composing with the projection G -> G/N — every element of G is told to behave according to the coset it lands in. This pullback turns cocycles on the quotient into cocycles on G and so induces a map on cohomology, in the opposite direction from restriction.
Precisely, let N be a normal subgroup of G and A a G-module. The fixed submodule A^N is a module over the quotient G/N. Inflation is the homomorphism inf : H^n(G/N, A^N) -> H^n(G, A) induced on cochains by the quotient map G -> G/N together with the inclusion A^N -> A. It is functorial and, like restriction, compatible with the long exact sequence and cup products.
Inflation and restriction fit into the inflation–restriction exact sequence, the simplest fragment of the Lyndon–Hochschild–Serre spectral sequence that computes H^*(G, A) from H^*(G/N, -) and H^*(N, -). In low degree it reads 0 -> H^1(G/N, A^N) -> H^1(G, A) -> H^1(N, A)^{G/N} -> H^2(G/N, A^N) -> H^2(G, A), exact, with the last map the transgression. This five-term sequence is one of the most-used computational tools in the subject.
The inflation–restriction sequence is a low-degree shadow of the Lyndon–Hochschild–Serre spectral sequence, which has E_2 page E_2^{p,q} = H^p(G/N, H^q(N, A)) converging to H^{p+q}(G, A). Inflation is the edge map into the base, restriction the edge map onto the fiber.