Hopfianness is not subgroup-closed

Statement
It is possible to have a Hopfian group $$G$$ and a subgroup $$H$$ of $$G$$ that is not Hopfian.

Stronger facts

 * Hopfianness is not characteristic subgroup-closed

Opposite facts

 * Hopfianness is direct factor-closed

Proof
Let $$G$$ be free group:F2 and $$H$$ be the derived subgroup of $$G$$.


 * $$G$$ is Hopfian because finitely generated and free implies Hopfian.
 * $$H$$ is not Hopfian: $$H$$ is a free group on a countable number of generators. It admits a surjective endomorphism that is not an automorphism obtained via a surjective map that is not bijective from the freely generating set for $$H$$ to itself.