Central factor is transitive

Statement
If $$H \le K \le G$$ are groups such that $$H$$ is a central factor of $$K$$ and $$K$$ is a central factor of $$G$$. Then, $$H$$ is a central factor of $$G$$.

Central factor
A subgroup $$H$$ of a group $$G$$ is termed a central factor of $$G$$ if every inner automorphism of $$G$$ restricts to an inner automorphism of $$H$$.

In terms of the function restriction expression, this is expressed as:

Inner automorphism $$\to$$ Inner automorphism

In other words, every inner automorphism of the whole group restricts to an inner automorphism of the subgroup.

Related facts

 * Central factor implies transitively normal
 * Central factor implies conjugacy-closed normal
 * Conjugacy-closed normality is transitive

Proof in terms of the function restriction expression
The property of being a central factor is a balanced subgroup property in terms of the function restriction formalism: it has a function restriction expression where both sides are equal (in this case, equal to the property of being an inner automorphism). Any balanced subgroup property is transitive, and thus, the property of being a central factor is transitive.

Hands-on proof
Given: Groups $$H \le K \le G$$ such that $$H$$ is a central factor of $$K$$ and $$K$$ is a central factor of $$G$$.

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

Proof: We need to show that any inner automorphism of $$G$$ restricts to an inner automorphism of $$H$$. Suppose $$\sigma$$ is an inner automorphism of $$G$$.

Since $$K$$ is a central factor of $$G$$, $$\sigma$$ restricts to an inner automorphism, say $$\sigma'$$, of $$K$$. Further, since $$H$$ is a central factor of $$K$$, $$\sigma'$$ restricts to an inner automorphism of $$H$$, say $$\sigma''$$.

Clearly, $$\sigma''$$ is also the restriction of $$\sigma$$ to $$H$$, so $$\sigma$$ restricts to an inner automorphism of $$H$$.