  A group G is (finitely) co-Hopfian if it does not contain any proper
(finite-index) subgroups isomorphic to itself. We study finitely generated
groups G that admit a descending chain of proper normal finite-index subgroups,
each of which is isomorphic to G. We prove that up to finite index, these are
always obtained by pulling back a chain of subgroups from a free abelian
quotient. We give two applications: First, we show any characteristic proper
finite-index subgroup isomorphic to G arises by pulling back a finite-index
subgroup of the abelianization, and secondly, we prove special cases (for
normal subgroups) of conjectures of Benjamini and Nekrashevych--Pete regarding
the classification of scale-invariant groups.
