Let $G$ be a group and $A$ be an abelian group. Let $\beta$, $\alpha :G\rightarrow Aut(A)$ be two homomorphisms. It is well known that if there exist $\sigma \in Aut(A)$, $\rho \in Aut(G)$ such that $(\beta \circ \rho )(g)=\sigma \circ \alpha (g)\circ \sigma^{-1}$ for all $g\in G$, then the semidirect products $A\rtimes _{\alpha }G$ and $A\rtimes_{\beta}G$ are isomorphic.
However, in one stage of a proof, I get that there exist $\sigma \in Aut(A)$, $\rho \in $ $Aut(G)$ such that $(\alpha \circ \rho )(g)=\sigma \circ \alpha (g)\circ \sigma^{-1}$ for all $g\in G$. Is there any interpretation of this formula in group theory? why this might be an interesting property?
Thank you in advance.
Not sure if this is what you see as interpretation, but rather something to note which I cannot write in a comment: You condition can be rephrased as the fact that the following diagram commutes:
$$\require{AMScd} \begin{CD} G @>{\rho}>> G\\ @VV{\alpha}V @VV{\alpha}V \\ \text{Aut}(A) @>{\text{conj}(\sigma)}>> \text{Aut}(A) \end{CD}$$
Here $\text{conj}: A \rightarrow \text{Aut}(A)$ is the group action of $A$ on itself via conjugation.