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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 Edit:It seems my information is somewhat out of date (see Milne's answer).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.
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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 Edit:It seems my information is somewhat out of date (see Milne's answer).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 $\frac{2}{27}2^{n^3 2^{\frac{2}{27}n^3 + O(n^{8/3})}$.

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$.
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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, $\overline{\operatorname{lim}} \, \frac{\log G(n)}{(\log n)^3} = \log 2$.