Hopfianness is not subgroup-closed

This article gives the statement, and possibly proof, of a group property (i.e., Hopfian group) not satisfying a group metaproperty (i.e., subgroup-closed group property).
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Statement

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

Related facts

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.