Normal closure of closed subgroup is closed in a f.g. profinite group?

Question

Is there an example of a topologically finitely generated profinite group $G$ and a closed subgroup $H$ such that we simultaneously have:

1. The profinite normal closure $L$ of $H$ is open in $G$.
2. The abstract normal closure $\Lambda$ of $H$ is not closed in $G$.

We clearly have that $\Lambda$ is dense in $L$, so (1) + (2) is equivalent by the result of Nikolov and Segal to:

1. The profinite normal closure $L$ of $H$ is open in $G$.
2. The abstract normal closure $\Lambda$ of $H$ has infinite index in $G$.

It seems to me that this should be impossible by a standard inverse limit argument, but I still couldn't quite put my finger on it.