 Finite metabelian groups are monomial

Let $L◃G$ 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 $\phi$. Every subgroup containing $L$ is normal. If an element $x\notin A$ stabilizes $\phi$, its action on the nonzero $\phi$-isotypic component has an eigenvector; that line is invariant under $⟨A,x⟩$ and affords a larger linear constituent, contradicting maximality. Thus $I_G(\phi )=A$, and the Clifford correspondence gives $\chi ={\phi }^G$.