Automorphism group action lemma for quotients

Statement
Suppose $$A$$ is an fact about::abelian normal subgroup of a group $$H$$, and $$\sigma'$$ is an automorphism of $$H$$ that restricts to an automorphism $$\alpha$$ of $$A$$. Note that this also shows that $$\sigma'$$ descends to an automorphism, say $$\sigma$$, of $$H/A$$.

Since $$A$$ is abelian, we have an action of the quotient group on it by conjugation (see quotient group acts on Abelian normal subgroup), giving a homomorphism:

$$\rho:H/A \to \operatorname{Aut}(A)$$.

The claim is that:

$$\rho \circ \sigma = c_\alpha \circ \rho$$.

where $$c_\alpha$$ denotes conjugation by $$\alpha$$ as an element of $$\operatorname{Aut}(A)$$.

Related facts

 * Automorphism group action lemma