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The main aim of this article is to study quantitative structure of finite simple exceptional groups $\F_4(2^n)$ with $n>1$. Here, we prove that the finite simple exceptional groups $\F_4(2^n)$, where $2^{4n}+1$ is a prime number with $n>1$ a power of $2$, can be uniquely determined by their orders and the set of the number of elements with the same order. In conclusion, we give a positive answer to J. G. Thompson's problem for finite simple exceptional groups $\F_4(2^n)$.
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