No nontrivial homomorphism from quotient group not implies characteristic

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 $$G$$.

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 particular example::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 particular example::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 $$H$$.