Left-transitively complemented normal implies characteristic

Statement with symbols
Suppose $$H$$ is a subgroup of a group $$K$$ such that whenever $$K$$ is a complemented normal subgroup of some group $$G$$, $$H$$ is also a complemented normal subgroup of $$G$$. Then, $$H$$ is a characteristic subgroup of $$K$$. In particular, $$H$$ is a fact about::complemented characteristic subgroup of $$K$$.

Related facts

 * Left transiter of normal is characteristic
 * Left residual of normal by complemented normal equals characteristic

Converse
Obviously, the naive converse is not true, since a characteristic subgroup need not be a complemented normal subgroup. However, even if we consider complemented characteristic subgroups, they need not be left-transitively complemented normal.

Facts used

 * 1) uses::Left residual of normal by complemented normal equals characteristic

Hands-on proof
Holomorph construction --

Proof using given facts
The proof follows directly from fact (1).