Let $L/K$ be a finite extension of number field.
Let $S$ be a set of places of $K$ containing Archimedean places. Let write $K_S$ for the largest subfield of $\overline{K}$ containing $K$ that is ramified only at primes in $S$. Let $S'$ be set of places of $L$ above $S$.
I think $K_S \subset L_{S'}$, but can we say the other inclusion $L_{S'} \subset K_{S}$in the case $L\subset K_S$?