If $F$ is a finite Galois extension over $K$ and $Gal(F/K)$ is abelian, then every subfield of $F$ containing $K$ is stable?

Question

I am studying Galois theory and ran across this problem.

If $F$ is a finite Galois extension over $K$ and $Gal(F/K)$ is abelian, then every subfield of $F$ containing $K$ is stable. Is the converse true?

How can I prove this? I guess the converse is false.