Generalized quaternion groups of order $4p$ are $4$-BCI-groups

For a finite group $G$ and a subset $S\subseteq G$, the bi-Cayley graph $\mathrm{BCay}(G,S)$ has vertex set $G\times \{0,1\}$ and edges $\{(x,0),(sx,1)\}$ for $x\in G$ and $s\in S$. The group $G$ is called an $m$-BCI-group if, for all subsets $S,T\subseteq G$ with $\mathrm{\mid }S\mathrm{\mid }\le m$, the isomorphism $\mathrm{BCay}(G,S)\cong \mathrm{BCay}(G,T)$ implies that $T=g{S}^{\alpha }$ for some $g\in G$ and $\alpha \in \mathrm{Aut}G$. We prove that, for every prime $p$, the generalized quaternion group $Q_{4p}$ is a $4$-BCI-group. For $p\ge 5$, the proof uses regular coverings and voltage assignments: we pass to quotient graphs modulo the Sylow $p$-subgroup and lift isomorphisms between them.