  Given a homeomorphism $T \colon X \to X$ of a compact metric space $X$, the
stabilized automorphism group $\textrm{Aut}^{\infty}(T)$ of the system $(X,T)$
is the group of self-homeomorphisms of $X$ which commute with some power of
$T$. We study the question of spatiality for stabilized automorphism groups of
shifts of finite type. We prove that any isomorphism $\Psi \colon
\textrm{Aut}^{\infty}(\sigma_{m}) \to \textrm{Aut}^{\infty}(\sigma_{n})$
between stabilized automorphism groups of full shifts is spatially induced by a
homeomorphism $\hat{\Psi}$ between respective stabilized spaces of chain
recurrent subshifts. This spatialization in particular gives a bijection
between the sets of periodic points which intertwines some powers of the
shifts, and this bijection recovers the isomorphism at the level of the
faithful actions on the sets of periodic points. We also prove that the outer
automorphism group of $\textrm{Aut}^{\infty}(\sigma_{n})$ is uncountable, and
deduce several other properties of $\textrm{Aut}^{\infty}(\sigma_{n})$ using
the spatiality results.
