I'm familiar with two primary ways to discuss the n$^{th}$ cohomology of groups with coefficients in an abelian group $A$:
(1) Through the exploration of n-fold extensions.
(2) By examining the map $d_n: C^n(G, A) \to C^{n+1}(G, A)$.
Many references I've encountered predominantly focus on the second approach for deriving cohomology. When it comes to utilizing the first method involving extensions, most discussions seem to be limited to n=2. I'm particularly interested in understanding the $\textbf{third cohomology}$ of a group via extensions. However, I'm struggling to find a comprehensive reference on this topic. Could you provide insights or direct me to appropriate $\textbf{references}$?