Let $H,K$ be closed subgroups of profinite group $G$ with $K\subset H$.
Then lemma says that if $H/K$ is finite then there exists a continuous section $s:G/H\rightarrow G/K$. Here, section means a map $s:G/H\rightarrow G/K$ whose composition with projection $G/K\rightarrow G/H$ is identity.
Existance of a section looks like obvious but I don't know how to handle the continuity.
Any idea or hint will be thankful.
I guess that it is enough to assume that $G$ is a topological group and $H,K$ are subgroups of $G$ to prove the lemma.