gr.group theory - to test equivalence of representations under automorphism

I assume that $phi$ is an automorphism of $G.$ Note that if $phi$ is inner then trivially $rho$ and $rhocircphi$ are equivalent, thus the answer depends only on the image of $phi$ in the outer automorphism group $Out(G).$

If $G$ is a finite group (or, more generally, compact group) and representations are finite-dimensional, so that they are determined up to isomorphism by their characters, then this problem admits a complete theoretical solution using the character theory. The automorphism group $Aut(G)$ acts on the set ${C_i}$ of the conjugacy classes of $G$, this action factors through the action of $Out(G),$ and

$$chi_{rhocircphi}(C)=chi_{rho}(phi(C)),qquad (*)$$

where $chi_rho$ is the character of $rho$ and $C$ is any conjugacy class. Since representations are determined by their characters,

$$rhocircphisimeqsigma iff chi_{rhocircphi}=chi_sigma,$$

which can be tested using $(*).$ Of course, applying this method in practice requires good knowledge of the character table of $G$ and the outer automorphism group $Out(G).$