Direct factor implies right-quotient-transitively central factor

Verbal statement
Any direct factor of a group is a right-quotient-transitively central factor.

Statement with symbols
Suppose $$G$$ is a group, $$H \le K \le G$$, and $$H$$ is a direct factor of $$G$$. Suppose, further, that $$K/H$$ is a central factor of $$G/H$$. Then, $$K$$ is a central factor of $$G$$.

Proof
Given: A group $$G$$, a direct factor $$H$$ of $$G$$, $$K$$ contains $$H$$ and $$K/H$$ is a central factor of $$G/H$$.

To prove: $$K$$ is a central factor of $$G$$.

Proof: Since $$H$$ is a direct factor of $$G$$, there exists a normal complement $$L$$ to $$H$$ in $$G$$, with $$HL = G$$ and $$H \cap L$$ trivial. Let $$M = K \cap L$$.

Consider the map $$\rho:L \to G/H$$ that sends every element of $$L$$ to its $$H$$-coset in $$G$$. This map is an isomorphism, since the kernel $$H \cap L$$ is trivial, and $$\rho^{-1}(K/H) = M$$. Thus, $$HM = K$$.


 * 1) Every element of $$H$$ centralizes every element of $$L$$: Since both $$H$$ and $$L$$ are normal in $$G$$, $$[H,L]$$ is contained in both, and since they intersect trivially, $$[H,L]$$ is trivial, so every element of $$H$$ centralizes every element of $$L$$.
 * 2) $$MC_L(M) = L$$: Since $$\rho$$ is an isomorphism, and $$\rho(M)$$ is a central factor of $$G/H$$, $$M$$ is a central factor of $$L$$.
 * 3) $$C_L(M)$$ centralizes $$K$$, i.e., $$C_L(M) \le C_G(K)$$: By step (1), $$C_L(M)$$ centralizes $$H$$, since $$L$$ centralizes $$H$$. It also centralizes $$M$$ by definition. Thus, it centralizes $$HM = K$$.
 * 4) $$KC_L(M) = G$$: We have $$KC_L(M) = HMC_L(M) = HL = G$$.
 * 5) $$KC_G(K) = G$$: This follows from the previous two steps.

This completes the proof.