On some special actions 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 concerning the representation of some special groups, 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.

It looks like you've let $\beta=\alpha$.  So of course this is true with $\sigma$ and $\rho$ the identity automorphisms.  But this is basically a trivial statement.
Not sure whether anything can be done with it if $\sigma\ne\rho$.  If there are any examples, I guess they could be called $\alpha$ -equivalent
