Let $l/k$ be a (may be infinite) galois extension with galois group $G$ and $k$ a finite field with size $q$. Also $k$ and $l$ are given the discrete topology. $G$ is given the Krull topology.
Then consider the Frobenius map $\varphi(\alpha)=\alpha^q$.
Let $a \in Z^1(G,l^*)$ be a 1-cocycle. Then let H be the subgroup Gal$(l/k(a(\varphi)))$ of $G$.
I have already proved that $a$ restricted to $H$ is a group homomorphism: $$a_{|H}:H \to k^*.$$
But I am stuck at proving that $a$ is continuous viewed without restriction: $$a:G \to l^*$$
So far I have that elements in the same coset of $H$ differ by an element from $k^*$ so $a$ has at most $q$ different values on a coset of $H$, so $a$ maps $G$ to a finite, hence compact, subset of $l^*$. Secondly that $a$ is constant on the orbits of the action of $H$ on $G$ by conjugation. So :
$$a(hgh^{-1})=a(g),\quad h\in H, g \in G.$$
I tried to prove from this that $a^{-1}(\{d\})$ has finite index in $G$ for any $d \in l^*$(EDIT: I have proved this.), but don't feel confident about this route anymore since this pre-image does not have to be subgroup.
Could someone point me in the right way? Or give me hints of any kind.