N-invariants send injective G-modules to Q-acyclics

Inflate a $Q$-module $V$ to $G$ by $gv=\pi (g)v$. This leaves the underlying groups and all arrows unchanged, hence preserves exact sequences. A $G$-map from an inflated $V$ to $I$ has image in $I^N$, since $N$ acts trivially on $V$. Conversely a $Q$-map $V\to I^N$ is a $G$-map after inclusion. These inverse correspondences prove the inflation–invariants adjunction directly.

For a $Q$-monomorphism $U\hookrightarrow V$ and map $U\to I^N$, inflate both and compose into $I$. By F2 the resulting $G$-map extends over $V$. Its image is $N$-fixed, so step 1.1 turns it back into a $Q$-map $V\to I^N$ extending the original. This is the defining injectivity property. It applies also to zero modules and to either trivial subgroup or quotient, and makes only one existential extension at a time.

Under DC, apply F3 to invariants for $Q$ and the injective $I^N$. It gives the asserted vanishing for all positive degrees; degree zero is $(I^N{)}^Q=I^G$, which need not vanish. For the relative branch, F4 identifies $R_J^p((-{)}^Q)(I^N)$ with the degree-$p$ cohomology after applying invariants to $J(I^N{)}_{\mathrm{d}\mathrm{e}\mathrm{l}}$. The supplied cochain-homotopy equivalence remains one after applying the additive invariants functor and compares this complex with $((I^N{)}^Q)[0]$, whose positive cohomology is zero. Thus the displayed relative vanishing follows without any choice. DC is used only for the resolution-independent group-cohomology conclusion, not step 2.1 or the explicitly supplied relative comparison.