The behaviour of $F(n)$ varies dramatically with the prime-factorization of $n$. Typically one gets much larger values for $F(n)$ when $n$ is the power of a prime, particularly when that prime is equal to $2$. The first key result $(\dagger)$ in this area (I believe) is due to Higman and Sims:
Theorem Let $p$ be a (fixed) prime number. Define $f(n,p)$ as the number of groups of order $p^n$. Then: $$f(n,p) = p^{(2/27 + o(1))n^3}.$$
(The link above gives a more detailed version of this result.) A result of Laci Pyber can be combined with that of Higman and Sims to give:
Theorem: Let $n=\prod_{i=1}^kp_i^{g_i}$ be a positive integer with the $p_i$ distinct primes. Let $\mu$ be the maximum of the $g_i$. The number of groups of order $n$ is at most $$n^{2/27+o(1)\mu^2}$$ as $\mu\to\infty$.
$(\dagger)$ I said this was a conjecture earlier - my mistake.