Let $G$ be a group and $\pi(G)$ be the set of primes $p$ such that $G$ contains an element of order $p$. Let $\operatorname{nse}(G)$ be the set of the numbers of elements of $G$ of the same order. We prove that the simple group $L_2(3^n)$ is uniquely determined by $\operatorname{nse}(L_2(3^n))$, where $|\pi(L_2(3^n))|=4$.