Quasirecognition by prime graph of the simple group $^2G_2(q)$

Let $G$ be a finite group. The main result of this paper is as follows: If $G$ is a finite group, such that $\Gamma(G)=\Gamma(^2G_2(q))$, where $q=3^{2n+1}$ for some $n\ge 1$, then $G$ has a (unique) nonabelian composition factor isomorphic to $^2G_2(q)$. We infer that if $G$ is a finite group satisfying $|G|=|^2G_2(q)|$ and $\Gamma(G)=\Gamma(^2G_2(q))$ then $G\cong{}^2G_2(q)$. This enables us to give new proofs for some theorems; e.g., a conjecture of W. Shi and J. Bi. Some applications of this result are also considered to the problem of recognition by element orders of finite groups.