Perfectness is not characteristic subgroup-closed

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 particular example::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.