We know that if $G$ is a profinite group and $H$ a closed subgroup of $G$ then we can write $$H = \varprojlim H/N$$ where $N$ varies on the open normal subgroups of $H$. Now, if $K$ is a closed subgroup of $H$, can we write $$H = \varprojlim \frac{H}{KN}$$ where $N$ varies on the open normal subgroups of $H$?
The question is motived by a problem that I'm trying to solve. I have a closed subgroup $K$ of $H$ and I would like to write $H$ as an inverse limit of quotient $H/U$ where each $U$ contains $K$, but I don't know if it is true.