Conjunction of normality with any nontrivially satisfied group property not implies characteristic

In terms of groups
Suppose $$H$$ is a nontrivial group. Then, there exists a nontrivial group $$G$$ containing $$H$$ as a normal subgroup such that $$H$$ is not a characteristic subgroup of $$G$$.

In terms of properties
Suppose $$\alpha$$ is a group property satisfied by some nontrivial group. Then, the property of being a normal subgroup that satisfies $$\alpha$$ as a group does not imply the property of being a characteristic subgroup.

Related facts

 * Weaker than::Every nontrivial normal subgroup is potentially normal-and-not-characteristic
 * Normal not implies characteristic in any nontrivial subquasivariety of the quasivariety of groups