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

Statement

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

• Every nontrivial normal subgroup is potentially normal-and-not-characteristic
• Normal not implies characteristic in any nontrivial subquasivariety of the quasivariety of groups