= Finite metabelian groups are monomial
Let $L\triangleleft G$ and both $L,G/L$ be abelian. For an irreducible $G$-module, choose a subgroup $A\supseteq L$ of maximal order whose restriction contains a <linear character> $\varphi$. Every subgroup containing $L$ is normal. If an element $x\notin A$ stabilizes $\varphi$, its action on the nonzero $\varphi$-<isotypic component> has an eigenvector; that line is invariant under $\langle A,x\rangle$ and affords a larger linear constituent, contradicting maximality. Thus $I_G(\varphi)=A$, and the <Clifford correspondence> gives $\chi=\varphi^G$.
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