Intermediate characteristicity is quotient-transitive

Statement with symbols
If $$H \le K \le G$$, with $$H$$ an intermediately characteristic subgroup of $$G$$, and $$K/H$$ an intermediately characteristic subgroup of $$G/H$$, then $$K$$ is an intermediately characteristic subgroup of $$G$$.

Facts used

 * 1) uses::Characteristicity is quotient-transitive
 * 2) uses::Intermediately operator preserves quotient-transitivity