Maximal abelian extension of finite abelian Galois extension of local fields of characteristic 0

Question

Let $K$ be a non-archimedean local field of characteristic $0$. One might take a finite extension of $\mathbb{Q}_p$ for some prime $p$. We fix an algebraic closure of $K$, and, throughout, every extension of $K$ is in that fixed algebraic closure. Let $L$ be a finite abelian Galois extension of $K$. I am wondering if there is any known relation between the maximal abelian extensions $L^{ab}$ of $L$ and $K^{ab}$ of $K$. For example, do we have $K^{ab}\subset L^{ab}$? If then, can we expect $\operatorname{Gal}(L/K)\simeq\operatorname{Gal}(L^{ab}/K^{ab})$?