Matrices of simple spectrum in irreducible representations of cyclic extensions of simple algebraic groups

Let $H$ be a linear algebraic group whose connected component $G\mathrm{\ne }1$ is simple and $H\mathrm{/}G$ is cyclic. We determine the irreducible projective representations $\varphi$ of $H$ such that $\varphi (G)$ is irreducible and $\varphi (h)$ has simple spectrum for some $h\in H$. The latter means that all irreducible constituents of the group $\varphi (⟨h⟩)$ are of multiplicity 1. (Here $⟨h⟩$ is the subgroup of $H$ generated by $h$.) This extends an earlier known result for $H=G$.