Let $k$ be a field, not necessarily algebraically closed, $G$ an affine group scheme over $k$, $H$ a subgroup of $G$, and $N$ a normal subgroup of $H$, none of them assumed to be smooth. Suppose that $g \in G(\overline k)$ is such that the image of $g$ in $(G/H)(\overline k)$ is contained in $(G/H)(k)$. Is $g N\_{\overline k} g^{-1}$ defined over $k$?


By Galois descent, it suffices to assume that $k$ is separably closed. Then, if $H$ is smooth, we have that the fibre of $G \to G/H$ over $g$ is smooth, hence contains a rational point, so that the desired result is clear. Thus the problem is only of interest if $H$ is neither smooth, nor even contained in a smooth subgroup of $\operatorname{Norm}\_G(N)$.



Fields of definition of conjugates
The answer is yes, and affineness is unnecessary. We'll assume that $G$ is locally of finite type (as I imagine was intended), but this is not essential. In brief, the proof proceeds by using fppf descent in place of Galois descent; this will involve working over a base slightly more general than a field (i.e., a finite algebra over a field). See Bosch–Lütkebohmert–Raynaud, *[Néron models](https://doi.org/10.1007/978-3-642-51438-8)*, 6.1/6 for the relevant descent statements. If $X$ is a $k$-scheme and $k'$ is a $k$-algebra, we will write $X\_{k'} = X \times\_{\operatorname{Spec} k} \operatorname{Spec} k'$. By SGA3, Exp. VI\_A, Thm. 3.2, the scheme quotient $G/H$ represents the quotient in the sense of fppf sheaves, so the meaning of $G/H$ is unambiguous.


By assumption, $g \in G(\overline{k})$ has image $[g]$ in $(G/H)(\overline{k})$ which lies in $(G/H)(k)$. Since $G$ is locally of finite type, there is a finite (not necessarily separable) extension $k'$ of $k$ such that $g \in G(k')$. Since $G/H$ is a sheaf for the fppf topology, the sequence of maps
$$
1 \to (G/H)(k) \to (G/H)(k') \rightrightarrows (G/H)(k' \otimes\_k k')
$$
is exact. In particular, if $i\_1, i\_2: k' \to k' \otimes\_k k'$ are the two natural ring maps, then $i\_1^\*[g] = i\_2^\*[g]$. Moreover, the fact that $G/H$ represents the fppf sheaf quotient implies that there is a faithfully flat, finite type ring map $f: k' \otimes\_k k' \to k''$ and $h'' \in H(k'')$ such that $f^\*i\_1^\*g \cdot h'' = f^\*i\_2^\*g$. Since $N$ is a $k$-scheme, we have $i\_1^\*N\_{k'} = i\_2^\*N\_{k'}$. We have $h''N\_{k''} (h'')^{-1} = N\_{k''}$ since $N$ is normal in $H$ by hypothesis, and thus
$$
(i\_1^\*g) N\_{k' \otimes\_k k'} (i\_1^\*g)^{-1} = (i\_2^\*g) N\_{k' \otimes\_k k'} (i\_2^\*g)^{-1}
$$
since this may be checked after pulling back via $f$. This means that $i\_1^\*(gN\_{k'}g^{-1}) = i\_2^\*(gN\_{k'}g^{-1})$, and thus by fppf descent for affine morphisms (e.g., closed immersions), we see that $gN\_{k'}g^{-1}$ is the base change to $k'$ of a closed subscheme of $G\_{k'}$.


