  For a homeomorphism $T \colon X \to X$ of a compact metric space $X$, the
stabilized automorphism group $\text{Aut}^{(\infty)}(T)$ consists of all
self-homeomorphisms of $X$ which commute with some power of $T$. Motivated by
the study of these groups in the context of shifts of finite type, we introduce
a certain entropy for groups called local $\mathcal{P}$ entropy. We show that
when $(X,T)$ is a non-trivial mixing shift of finite type, the local
$\mathcal{P}$ entropy of the group $\text{Aut}^{(\infty)}(T)$ is determined by
the topological entropy of $(X,T)$. We use this to give a complete
classification of the isomorphism type of the stabilized automorphism groups of
full shifts.
