Distinguishing Gauss sums

Let $\F_q$ be the field of order $q = p^f$ for prime $p$. If $G(χ)$ is the Gauss sum attached to a multiplicative character $χ$ on $\F_q^\times$, then $G(χ) = G(χ^p)$. We investigate the converse question: when does the equality of Gauss sums $G(χ_2) = G(χ_1)$ imply that $χ_2 = χ_1^{p^j}$ for some i