A group extension of an abelian group $H$ by an abelian group $G$ consists of a group $E$ together with an inclusion of $G \hookrightarrow E$ as a normal subgroup and a surjective homomorphism $E \twoheadrightarrow H$ that displays $H$ as the quotient group $E/G$. This data is typically displayed in a diagram of group homomorphisms: \(0 \to G \to E \to H \to 0.\) A pair of group extensions $E$ and $E’$ of $G$ and $H$ are considered to be equivalent whenever there is an isomorphism $E \cong E’$ that \emph{commutes with} the inclusions of $G$ and quotient maps to $H$, in a sense that is made precise in \S\ref{sec:diagram-chase}. The set of equivalence classes of \emph{abelian} group extensions $E$ of $H$ by $G$ defines an abelian group $\mathrm{Ext}(H,G)$.