Perfectness is not characteristic subgroup-closed

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

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

Related facts

• Perfectness is not subgroup-closed
• Every group is a subgroup of a perfect group
• Every finite group is a subgroup of a finite perfect group

Proof

• Let $G$ be special linear group:SL(2,5).
• Let $H$ be the center of special linear group:SL(2,5), i.e., the center of $G$. Since center is characteristic, $H$ is characteristic in $G$.
• $H$ is not an abelian group.