  Let $k = \mathbb{F}_p$ or $\mathbb{Z}_p$ (or finite extensions of these). Let
$G$ be a $p$-valuable group, and form its completed group algebra $kG$. By
analysing the conjugation action of $G$ on itself, we prove two structural
results. Firstly, we show that all inner automorphisms of $kG$ that preserve
$G$ are induced from inner automorphisms of $G$. Secondly, for a closed
subgroup $\Gamma$ of $G$, we calculate the $\Gamma$-fixed ring of $kG/I$ under
the conjugation action of $\Gamma$, for certain ideals $I$ induced from the
$G$-centraliser of $\Gamma$.
