Pure definability is quotient-transitive

Statement with symbols
Suppose $$H \le K \le G$$ are groups such that $$H$$ is a purely definable subgroup of $$G$$ and $$K/H$$ is a purely definable subgroup of $$G/H$$ (note that any purely definable subgroup is characteristic and hence normal, so it makes sense to take the quotient group). Then, $$K$$ is a purely definable subgroup of $$G$$.