Are there some good asymptotic estimations for the number $F(n)$ of nonisomorphic finite groups of size smaller than $n$?

The behaviour of $F(n)$ varies dramatically with the primefactorization of $n$. Typically one gets a large jump in the value of $F(n)$ as $n$ passes 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:
(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:
The best way in to this area (it seems to me) is to consult Pyber's paper on the subject containing the above result:
An interesting extra tidbit from that paper is the following:
(In other words all counts are dominated by $p$groups.) You should also refer to Derek's answer  apologies to him for not referencing his result! $(\dagger)$ I said this was a conjecture earlier  my mistake. 


It is proved in Holt, D. F., Enumerating perfect groups. J. London Math. Soc. (2) 39 (1989), no. 1, 67–78 that $n^{2l(n)^2/27−dl(n)} \le F(n) \le n^{l(n)^2/6+l(n)}$ for some constant $d$, where $l(n) = \log_2(n)$. The lower bound is coming from Higman's construction of large numbers of $p$groups of class 2, and the general belief seems to be that the lower bound is close to being the correct number. I think that there might be better results known now. You could try searching publications of Laszlo Pyber, but I don't have time right now! Added later: Nick's answer is much more accurate than mine. But, in case it is of any interest, let me add that the main result of the paper I mentioned was an estimate of the number ${\rm perf}(n)$ of finite perfect groups of order at most $n$, which (perhaps surprisingly) is also large, and satisfies: $n^{l(n)^2/108−cl(n)} \le {\rm perf}(n) \le n^{l(n)^2/48+l(n)}$, where $c=11/36$. 

