  A group $\Gamma$ is said to be finitely non-co-Hopfian, or renormalizable, if
there exists a self-embedding $\varphi \colon \Gamma \to \Gamma$ whose image is
a proper subgroup of finite index. Such a proper self-embedding is called a
renormalization for $\Gamma$. In this work, we associate a dynamical system to
a renormalization $\varphi$ of $\Gamma$. The discriminant invariant ${\mathcal
D}_{\varphi}$ of the associated Cantor dynamical system is a profinite group
which is a measure of the asymmetries of the dynamical system. If ${\mathcal
D}_{\varphi}$ is a finite group for some renormalization, we show that
$\Gamma/C_{\varphi}$ is virtually nilpotent, where $C_{\varphi}$ is the kernel
of the action map. We introduce the notion of a (virtually) renormalizable
Cantor action, and show that the action associated to a renormalizable group is
virtually renormalizable. We study the properties of virtually renormalizable
Cantor actions, and show that virtual renormalizability is an invariant of
continuous orbit equivalence. Moreover, the discriminant invariant of a
renormalizable Cantor action is an invariant of continuous orbit equivalence.
Finally, the notion of a renormalizable Cantor action is related to the notion
of a self-replicating group of automorphisms of a rooted tree.
