On $\text{BCI}$-groups and $\text{CI}$-groups

Let $G$ be a finite group and $S$ be a subset of $G\mathrm{.}$ A bi-Cayley graph $\text{BCay}(G,S)$ is a simple and an undirected graph with vertex-set $G\times \{1,2\}$ and edge-set $\{\{(g,1),(sg,2)\}\mid g\in G,s\in S\}$. A bi-Cayley graph $\text{BCay}(G,S)$ is called a $\text{BCI}$-graph if for any bi-Cayley graph $\text{BCay}(G,T)$, whenever $\text{BCay}(G,S)\cong \text{BCay}(G,T)$ we have $T=g{S}^{\sigma }$ for some $g\in G$ and $\sigma \in \text{Aut}(G)\mathrm{.}$ A group $G$ is called a $\text{BCI}$-group if every bi-Cayley graph of $G$ is a $\text{BCI}$-graph. In this paper, we showed that every $\text{BCI}$-group is a $\text{CI}$-group, which gives a positive answer to a conjecture proposed by Arezoomand and Taeri in \cite{arezoomand1}. Also we proved that there is no any non-Abelian $4$-$\text{BCI}$-simple group. In addition all $\text{BCI}$-groups of order $2p$, $p$ a prime, are characterized.