the derived length of $M_G(L_{n-2})$ is $n-1$ and thus, by the induction hypothesis, $M_G(L_{n-2})$ is an IYB group. By [@CJR Theorem 3.3], $G$ is an IYB group.

Our aim is to study the structure of finite left braces whose multiplicative groups are metabelian $A$-groups, with an expectation that under this natural restriction one can get decisive results.

For the convenience of the reader we recall some basic definitions and facts that we will use (see [@B3; @BCJO; @CJO2]). Let $(B,+,\cdot)$ be a left brace. The lambda map of $B$ is the homomorphism $\lambda\colon (B,\cdot)\longrightarrow {\operatorname{Aut}}(B,+)$ defined by $\lambda(a)=\lambda_a$ and $\lambda_a(b)=ab-a$ for all $a,b\in B$. Recall that a left ideal of the left brace $B$ is a subgroup $L$ of its additive group such that $\lambda_b(a)\in L$ for all $a\in L$ and all $b\in B$. If $L$ is a left ideal of $B$, then $b^{-1}c=\lambda_{b^{-1}}(c-b)\in L$, for all $b,c\in L$, thus $L$ also is a subgroup of the multiplicative group of $B$. An ideal of $B$ is a left ideal $I$ of $B$ such that $I$ is a normal subgroup of $(B,\cdot)$; in fact, ideals of left braces are precisely the kernels of left brace homomorphisms. A nonzero left brace $B$ 