I am reading Class field theory - Bonn Lectures by Neukirch.

Given a $G$ module $A$ he defines Cohomology groups $H^i(G,A) : i\in \mathbb{Z}$ by considering some complete resolution of $G$ modules and applying hom functor.

He then calculated $H^{-1}(G,A),H^0(G,A),H^1(G,A),H^2(G,A)$

It says $H^2(G,A)$ is related to group extensions. Given an abelian group $A$ written multiplicatively and an arbitrary subgroup $G$, we want to find all group extensions $\hat{G}$ of $A$ such that $A$ is normal subgroup pf $\hat{G}$ and $\hat{G}/A\cong G$. He says this is related to $H^2(G,A)$.

It says if the group $G$ acts trivially on $A$ then $H^1(G,A)=\rm{Hom}(G,A)$, in particular for $A=\mathbb{Q}/\mathbb{Z}$ we have character group $H^1(G,A)=\chi(G)$.

It says $H^0(G,A)=A^G/N_GA$ the norm residue group of $G$ module $A$ with notation $A^G=\{a\in A : \sigma a=a~\forall \sigma\in G\}$ and $N_GA=\{\sum_{g\in G}ga:a\in A\}$. It says this is useful in class field theory.

It says $H^{-1}(G,A)$ has some other concrete form.

It says $H^{-2}(G,\mathbb{Z})=G^{ab}$ abelianization of $G$.

My question is are there any higher cohomology groups that are of interest at least in some special cases?