Lifting representations of finite reductive groups II: Explicit conorms

Let $k$ be a field, $\tilde{G}$ a connected reductive $k$-group, and $\Gamma$ a finite group. In a previous work, the authors defined what it means for a connected reductive $k$-group $G$ to be "parascopic" for $(\tilde{G},\Gamma)$. Roughly, this is a simultaneous generalization of several settings. For example, $\Gamma$ could act on $\tilde{G}$, and $G$ could be the connected part of the group of $\Gamma$-fixed points in $\tilde{G}$. Or $G$ could be an endoscopic group, a pseudo-Levi subgroup, or an isogenous image of $\tilde{G}$. If $G$ is such a group, and both $\tilde{G}$ and $G$ are $k$-quasisplit, then we constructed a map $\hat{\mathcal{N}}^{\text{st}}$ from the set of stable semisimple conjugacy classes in the dual $G^\wedge(k)$ to the set of such classes in $\tilde{G}^\wedge(k)$. When $k$ is finite, this implies a lifting from packets of representations of $G(k)$ to those of $\tilde{G}(k)$. In order to understand such a lifting better, here we describe two ways in which $\hat{\mathcal{N}}^{\text{st}}$ can be made more explicit. First, we can express our map in the general case in terms of simpler cases. We do so by showing that $\hat{\mathcal{N}}^{\text{st}}$ is compatible with isogenies and with Weil restriction, and also by expressing it as a composition of simpler maps. Second, in many cases we can construct an explicit $k$-morphism $\hat N \colon G^\wedge \longrightarrow \tilde{G}^\wedge$ that agrees with $\hat{\mathcal{N}}^{\text{st}}$. As a consequence, our lifting of representations is seen to coincide with Shintani lifting in some important cases.