On Sylow numbers of some finite groups

Let $G$ be a finite group, let $\pi (G)$ be the set of primes $p$ such that $G$ contains an element of order $p$, and let $n_{p}(G)$ be the number of Sylow $p$-subgroups of $G$, that is, $n_{p}(G)=|\mathrm{Syl}_{p}(G)|$. Set $\mathrm{NS} (G):=\{n_{p}(G)|~p\in \pi (G)\}$. In this paper, we will show that if $ |G|=|S| $ and $\mathrm{NS}(G)=\mathrm{NS}(S)$, where $S$ is one of the groups: the special projective linear groups $L_{3}(q)$, with $5\nmid (q-1)$, the projective special unitary groups $U_{3}(q)$, the sporadic simple groups, the alternating simple groups, and the symmetric groups of degree prime $r$, then $G$ is isomorphic to $S$. Furthermore, we will show that if $G$ is a finite centerless group and $\mathrm{NS}(G)=\mathrm{NS}(L_{2}(17))$, then $G$ is isomorphic to $L_{2}(17)$, and or $G$ is isomorphic to $\mathrm{Aut}(L_{2}(17)$.