= Solution
The <commutator subgroup>, or derived subgroup, is
$$
G'=\langle[x,y]:x,y\in G\rangle,
\qquad [x,y]=x^{-1}y^{-1}xy.
$$
A <linear character> is afforded by a one-dimensional representation $\rho:G\to\mathbb C^\times$. Since $\mathbb C^\times$ is abelian, every commutator lies in $\ker\rho$, so
$$
\boxed{\ G'\leq\ker\chi.\ }
$$
Let $\pi:G\to G/G'$ be the quotient map. Every <irreducible representation of a finite abelian group> is one-dimensional, so each irreducible character $\psi$ of the <abelianization> $G/G'$ inflates to the linear character $\psi\circ\pi$ of $G$. Conversely, the containment $G'\leq\ker\chi$ makes every linear character of $G$ factor uniquely through $G/G'$. Thus <inflation of a group representation> gives a bijection
$$
\boxed{\ \operatorname{Irr}(G/G')\longleftrightarrow
\{\text{linear characters of }G\}.\ }
$$
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