No nontrivial homomorphism from quotient group not implies characteristic

This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., normal subgroup having no nontrivial homomorphism from its quotient group) need not satisfy the second subgroup property (i.e., characteristic subgroup)
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Statement

It is possible to have a group $G$ and a subgroup $H$ such that $H$ is a normal subgroup having no nontrivial homomorphism from its quotient group (i.e., there is no nontrivial homomorphism from $G/H$ to $H$) but is not a characteristic subgroup of $G$.

Proof

Infinite abelian example

Let $G$ be the additive group of rational numbers $\mathbb {Q}$ and $H$ be the subgroup $\mathbb {Z}$. Then:

• There is no nontrivial homomorphism from $G/H$ to $H$: This is because every element of $G/H$ has finite order, and no non-identity element of $H$ has finite order.
• $H$ is not characteristic in $G$: The automorphism that sends every element to its half in $\mathbb {Q}$ does not preserve $H$.

Finite example

Let $G$ be direct product of SL(2,3) and Z2 and $H$ be the second direct factor cyclic group:Z2. Then:

• There is no nontrivial homomorphism from $G/H$ to $H$: This is because the quotient group special linear group:SL(2,3) has no subgroup of index two. See subgroup structure of special linear group:SL(2,3).
• $H$ is not characteristic in $G$: There is an automorphism that keeps the first direct factor intact and replaces $H$ by the subgroup comprising the identity element and the product of the generator of $H$ with the non-identity central element of the first direct factor. This automorphism does not preserve $H$.