Characteristically complemented characteristic is transitive

Statement with symbols
Suppose $$H \le K \le G$$ are groups such that $$H$$ is a characteristically complemented characteristic subgroup of $$K$$ and $$K$$ is a characteristically complemented characteristic subgroup of $$G$$. Then, $$H$$ is a characteristically complemented characteristic subgroup of $$G$$.

Similar facts

 * Characteristicity is transitive
 * Retract is transitive
 * Direct factor is transitive

Opposite facts

 * Permutably complemented is not transitive
 * Lattice-complemented is not transitive
 * Complemented normal is not transitive
 * Complemented characteristic not implies left-transitively complemented normal

Facts used

 * 1) uses::Characteristicity is transitive
 * 2) uses::Characteristicity is strongly join-closed