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The asymptotics for groups are strictly speaking still open, since extensions of nonsolvable groups are apparently rather thorny. Edit:It seems my information is somewhat out of date (see Milne's answer). I'm not sure how bad the $o(1)$ can be.

It is expected that 2-groups dominate by a lot, although one could reasonably argue that the numerical evidence gathered to date samples the very small end of the nonsolvable family. By a 1965 result of Higman and Sims, the number of isomorphism types of groups of order $2^n$ (and conjecturally, groups of order at most $2^n$) grows as $2^{\frac{2}{27}n^3 + O(n^{8/3})}$.

In other words, your function $G(n)$ grows very roughly like $2^{\frac{2}{27}(\log_2 n)^3}$. More specifically, $\overline{\operatorname{lim}} \, \frac{\log G(n)}{(\log_2 n)^3} = 2/27$.

Addendum: I did a bit of GAP computation following Brian Conrad's comment. If we weight by dividing by the order of the automorphism group, none of the orders up to 70 contribute more than 1 (including 64, which contributes 48611383/78744960), and the average contribution from non-highly-divisible orders drops pretty quickly. The cumulative sums by 10s are roughly: 0, 5.3, 7.5, 8.9, 10.3, 11.4, 12.1, 13.1. Due to the jumpiness, I can only say that the growth is looks very sub-linear. Given the growth rate of isomorphism types, and possibly only a bit faster than a multiple I suspect we'll eventually get an explosion of logmass for large powers of two even with the weighting.

The asymptotics for groups are strictly speaking still open, since extensions of nonsolvable groups are apparently rather thorny. Edit:It seems my information is somewhat out of date (see Milne's answer). I'm not sure how bad the $o(1)$ can be.
It is expected that 2-groups dominate by a lot, although one could reasonably argue that the numerical evidence gathered to date samples the very small end of the nonsolvable family. By a 1965 result of Higman and Sims, the number of isomorphism types of groups of order $2^n$ (and conjecturally, groups of order at most $2^n$) grows as $\frac{2}{27}2^{n^3 2^{\frac{2}{27}n^3 + O(n^{8/3})}$.
In other words, your function $G(n)$ grows very roughly like $2^{(\log 2^{\frac{2}{27}(\log_2 n)^3}$. More specifically, $\overline{\operatorname{lim}} \, \frac{\log G(n)}{(\log G(n)}{(\log_2 n)^3} = \2/27$.
Addendum: I did a bit of GAP computation following Brian Conrad's comment. If we weight by dividing by the order of the automorphism group, none of the orders up to 70 contribute more than 1 (including 64, which contributes 48611383/78744960), and the average contribution from non-highly-divisible orders drops pretty quickly. The cumulative sums by 10s are roughly: 0, 5.3, 7.5, 8.9, 10.3, 11.4, 12.1, 13.1. Due to the jumpiness, I can only say that the growth is very sub-linear, and possibly only a bit faster than a multiple of log2$. 1 The asymptotics for groups are strictly speaking still open, since extensions of nonsolvable groups are apparently rather thorny. It is expected that 2-groups dominate by a lot, although one could reasonably argue that the numerical evidence gathered to date samples the very small end of the nonsolvable family. By a 1965 result of Higman and Sims, the number of isomorphism types of groups of order$2^n$(and conjecturally, groups of order at most$2^n$) grows as$\frac{2}{27}2^{n^3 + O(n^{8/3})}$. In other words, your function$G(n)$grows very roughly like$2^{(\log n)^3}$. More specifically, $\overline{\operatorname{lim}} \, \frac{\log G(n)}{(\log n)^3} = \log 2\$.