LHS defines low-degree group cohomology

For a supplied resolution $M\to I$, F1 defines $H^1(G,M)=\ker ((I^1{)}^G\to (I^2{)}^G)\mathrm{/}\mathrm{im}((I^0{)}^G\to (I^1{)}^G)$ and similarly uses degrees one, two and three for $H^2$. No group extension occurs in either definition. DC supplies the stated resolution-independent notation, while the displayed quotient for specified data is already defined in ZF. F2 uses these derived functors to identify both its page and its target.

The trivial extension $1\to 1\to G\stackrel{1}{\to }G\to 1$ makes the proposed independent definition visibly circular. Trivial-group invariants are the identity functor, so applied to an exact resolution their positive cohomology is zero and their degree-zero cohomology is $M$. The LHS second page is therefore $E_2^{p,0}=H^p(G,M)$ and zero elsewhere; its target is the same $H^p(G,M)$. In particular the entries at $(1,0)$ and $(2,0)$ already contain the groups allegedly being defined. LHS supplies a computation and comparison tool, not the missing definition. For zero coefficients all displayed quotients are zero, with the same dependency order.