Homomorph-containment satisfies intermediate subgroup condition

Verbal statement
A homomorph-containing subgroup of the whole group is also a homomorph-containing subgroup of any intermediate subgroup.

Statement with symbols
Suppose $$H \le K \le G$$ are groups, and $$H$$ is a homomorph-containing subgroup of $$G$$. Then, $$H$$ is also a homomorph-containing subgroup of $$K$$.

Related facts

 * Full invariance does not satisfy intermediate subgroup condition
 * Homomorph-containing implies intermediately fully invariant
 * Homomorph-containment does not satisfy image condition

Proof
Given: Groups $$H \le K \le G$$ such that $$H$$ is homomorph-containing in $$G$$. A homomorphism $$\varphi:H \to K$$.

To prove: $$\varphi(H)$$ is contained in $$H$$.

Proof: Since $$K \le G$$, we can compose $$\varphi$$ with the inclusion of $$K$$ in $$G$$ to get a homomorphism $$\varphi':H \to G$$. Since $$H$$ is homomorph-containing in $$G$$, $$\varphi'(H) \le H$$, so $$\varphi(H) \le H$$.