Generalized Quaternion Groups with the $m$-DCI Property

A Cayley digraph $\mathrm{Cay}(G,S)$ of a finite group $G$ with respect to a subset $S$ of $G$ is said to be a CI-digraph if for every Cayley digraph $\mathrm{Cay}(G,T)$ isomorphic to $\mathrm{Cay}(G,S)$, there exists an automorphism $\sigma$ of $G$ such that $S^\sigma=T$. A finite group $G$ is said to have the $m$-DCI property for some positive integer $m$ if every Cayley digraph $\mathrm{Cay}(G,S)$ of $G$ with $|S|=m$ is a CI-digraph, and is said to be a DCI-group if $G$ has the $m$-DCI property for all $1\leq m\leq |G|$. Let $\mathrm{Q}_{4n}$ be a generalized quaternion group (also called dicyclic group) of order $4n$ with an integer $n\geq 3$, and let $\mathrm{Q}_{4n}$ have the $m$-DCI property for some $1 \leq m\leq 2n-1$. It is shown in this paper that $n$ is odd, and $n$ is not divisible by $p^2$ for any prime $p\leq m-1$. Furthermore, if $n\geq 3$ is a power of a prime $p$, then $\mathrm{Q}_{4n}$ has the $m$-DCI property if and only if $p$ is odd, and either $n=p$ or $1\leq m\leq p$.