Classification of finite simple groups $G$ with $\pi(G)=\{2,3,5\}$

Question

Let $G$ be a finite group. The set of prime divisors of $|G|$ is denoted by $\pi(G)$. I am looking for the classification of finite simple groups $G$ with $\pi(G)=\{2,3,5\}$.

Answer 1

EDIT: Answered too quickly at first, improved answer now.

See this answer, which is very thorough: https://mathoverflow.net/questions/67671/number-of-prime-factors-of-the-order-of-a-finite-non-abelian-simple-group

Based on the paper of Bugeaud, Cao and Mignotte, the groups you are looking for are $A_5$, $A_6$, $PSp(4,3)$ and $PSU(4,2)$.

The literature on this seems a bit confusing to me. For example, I can't understand why $PSU(4,2)$ doesn't appear in Corollary of Leon&Wales.

Also, B. C. and M. say that the case of three primes was done by Herzog, but when I chase up that paper, there is additional hypothesis.