Let $L/K$ be a Galois field extension and $(I, \leq)$ be the directed set of all finite Galois extensions $E$ of $K$ contained in $L$ (we say $E^{\prime}\leq E$ if $E^{\prime}\subseteq E$). If $E^{\prime}\leq E$ we have a homomorphism $${\rm Gal}(E/K)\to {\rm Gal}(E^{\prime}/K)$$ given by the obvious restriction. These give rise to an inverse system so we obtain the inverse limit $\Lambda:= \varprojlim {\rm Gal}(E/K)$.
We have a restriction homomorphism $G:={\rm Gal}(L/K) \to {\rm Gal}(E/K)$ for each $E\in I$. These combine to give a group isomorphism $$\theta: G\cong \Lambda.$$
I need to show that $\theta^{-1}$ is continuous. Here, I assume $G$ has the Krull topology and $\Lambda$ the product topology where each component of $\Lambda$ is discrete.
Question: I want to show that it's enough to assume that $H$ is an open, normal subgroup of $G$. How can I show this, e.g. from the definition of the fundamental system of neighborhoods of the identity?
Many thanks in advance.