Let $k$ be a local field (if necessary, assume characteristic zero). In general, if $X$ is a smooth variety of finite type over $k$ of dimension $n$, then the set of $k$-rational points $X(k)$ is an analytic manifold over $k$ of dimension $n$. I was thinking about the passage $X \mapsto X(k)$ from smooth varieties to manifolds in the context of coset spaces, and had a couple of questions.

Let $H$ be a closed subgroup of a linear algebraic group $G$ over $k$. Assume $H$ is defined over $k$. Then the coset space $G/H$ has the structure of a smooth quasi-projective variety over $k$, which tells us that $G/H(k)$ is an analytic manifold over $k$ of dimension $n = \operatorname{Dim}G - \operatorname{Dim}H$.

On the other hand, $H(k)$ is a closed subgroup of the analytic Lie group $G(k)$, and the quotient $G(k)/H(k)$ has the structure of an $n$ dimensional analytic manifold (the quotient map $G(k) \rightarrow G(k)/H(k)$ is a principal fiber bundle with $H(k)$ as a fiber). This just follows from the general theory of Lie groups.

Of course, $G(k)/H(k)$ injects into $G/H(k)$, but they don't have to be equal. They are the same if $H^1(\operatorname{Gal}(k_s/k),H) $ is trivial.

What can we say about the inclusion $G(k)/H(k) \rightarrow G/H(k)$ from the perspective of manifolds? Is the inclusion an analytic map? Does it make $G(k)/H(k)$ into an open submanifold? How different can these two manifolds be?