Equivariant $v_{1,\vec{0}}$-self maps

Let $G$ be a cyclic $p$-group or generalized quaternion group, $X\in {\pi }_0{S}_G$ be a virtual $G$-set, and $V$ be a fixed point free complex $G$-representation. Under conditions depending on the sizes of $G$, $X$, and $V$, we construct a self map $v\text{\hspace{0.17em}⁣}:{\mathrm{\Sigma }}^{V}C(X{)}_{(p)}\to C(X{)}_{(p)}$ on the cofiber of $X$ which induces an equivalence in $G$-equivariant $K$-theory. These are transchromatic $v_{1,\vec{0}}$-self maps, in the sense that they are lifts of classical $v_1$-self maps for which the telescope $C(X{)}_{(p)}[v^{-1}]$ can have nonzero rational geometric fixed points.