Question:Normal subgroup characteristic in bigger group

Q: '''Suppose $$H$$ is a normal subgroup of a group $$G$$. Is it always possible to find a group $$K$$ containing $$G$$ such that $$H$$ is a characteristic subgroup of $$K$$?'''

A: Yes, this is always possible. Moreover, being normal is precisely the necessary and sufficient condition. The necessity of being normal follows from the fact that characteristic implies normal and normality satisfies intermediate subgroup condition. The sufficiency (i.e., that being normal guarantees the existence of a possible overgroup) is trickier. A full proof of the fact is available at the page normal equals potentially characteristic. Other stronger versions and related results are linked to from that page.

More basic: Question:Normal subgroup characteristic subgroup relation.