  Let $\mathcal{G}$ be a connected reductive almost simple group over the Witt
ring $W(\mathbb{F})$ for $\mathbb{F}$ a finite field of characteristic $p$. Let
$R$ and $R'$ be complete noetherian local $W(\mathbb{F})$ -algebras with
residue field $\mathbb{F}$. Under a mild condition on $p$ in relation to
structural constants of $\mathcal{G}$, we show the following results: (1) Every
closed subgroup $H$ of $\mathcal{G}(R)$ with full residual image
$\mathcal{G}(\mathbb{F})$ is a conjugate of a group $\mathcal{G}(A)$ for
$A\subset R$ a closed subring that is local and has residue field $\mathbb{F}$
. (2) Every surjective homomorphism $\mathcal{G}(R)\to\mathcal{G}(R')$ is, up
to conjugation, induced from a ring homomorphism $R\to R'$. (3) The identity
map on $\mathcal{G}(R)$ represents the universal deformation of the
representation of the profinite group $\mathcal{G}(R)$ given by the reduction
map $\mathcal{G}(R)\to\mathcal{G}(\mathbb{F})$. This generalizes results of
Dorobisz and Eardley-Manoharmayum and of Manoharmayum, and in addition provides
an abstract classification result for closed subgroups of $\mathcal{G}(R)$ with
residually full image.
  We provide an axiomatic framework to study this type of question, also for
slightly more general $\mathcal{G}$, and we study in the case at hand in great
detail what conditions on $\mathbb{F}$ or on $p$ in relation to $\mathcal{G}$
are necessary for the above results to hold.
