  Given a connected reductive group $\tilde{G}$ over a finite field $k$, and a
semisimple $k$-automorphism $\varepsilon$ of $\tilde{G}$ of finite order, let
$G$ denote the connected part of the group of $\varepsilon$-fixed points. Then
there exists a lifting from packets of representations of $G(k)$ to packets for
$\tilde{G}(k)$. In the case of Deligne-Lusztig representations, we show that
this lifting satisfies a character relation analogous to that of Shintani.
