Endomorphism kernel is quotient-transitive

Statement
Suppose $$H \le K \le G$$ are groups such that $$H$$ is an endomorphism kernel in $$G$$ and the quotient group $$K/H$$ is an endomorphism kernel in $$G/H$$. (Note that it makes sense to take quotients because endomorphism kernel implies normal). Then, $$K$$ is an endomorphism kernel in $$G$$.

Similar facts about similar properties

 * Normality is quotient-transitive
 * Characteristicity is quotient-transitive
 * Complemented normal is quotient-transitive

Opposite facts about endomorphism kernel

 * Endomorphism kernel is not finite-intersection-closed
 * Endomorphism kernel is not transitive

Definitions used
We use the following definition of endomorphism kernel: a normal subgroup $$A$$ of a group $$B$$ is an endomorphism kernel if there exists a subgroup of $$B$$ isomorphic to the quotient group $$B/A$$.

Facts used

 * 1) uses::Third isomorphism theorem: This basically tells us that $$(G/H)/(K/H) \cong G/K$$.

Proof
Given: Groups $$H \le K \le G$$ such that $$H$$ is an endomorphism kernel in $$G$$ and $$K/H$$ is an endomorphism kernel in $$G/H$$.

To prove: $$K$$ is an endomorphism kernel in $$G$$.

Proof: