Group with finitely many homomorphisms to any finite group

Equivalence of definitions
The nontrivial ingredients in the equivalences are Poincare's theorem (which asserts that a subgroup of finite index $$n$$ contains a normal subgroup of finite index at most $$n!$$) and that index satisfies intersection inequality, which yields that the intersection of finitely many subgroups of finite index again has finite index. We also need to use the fact that a subgroup of finite index can be contained in only finitely many intermediate subgroups.

Facts

 * Residually finite and finitely many homomorphisms to any finite group implies Hopfian
 * Finitely many homomorphisms to any finite group implies every subgroup of finite index has finitely many automorphic subgroups