Let $H$ be an open subgroup of a locally compact Hausdorff abelian group $G$. Assume that $G/H$ is a finitely generated abelian group. Let $\chi: H \rightarrow \mathbb{C}^{\ast}$ be a continuous homomorphism. Does $\chi$ extend to a continuous homomorphism into $\mathbb{C}^{\ast}$ defined on all of $G$?
If $\chi$ maps $H$ into the circle $S^1$, then $\chi$ does extend to a continuous homomorphism on all of $G$, also mapping into $S^1$. This follows from Pontryagin duality, and in fact this is true when $H$ is a closed, not necessarily open subgroup, and with no assumption about $G/H$.