Lubotzky proved that the number of $2$-generator groups of order at most $n$ (or order exactly $n$) is bounded by $n^{A \log n}$ for some constant $A$. The number of $2$ groups of order $2^m$ with $n \ge 2^m > n/2$ is (<b>corrected, see Will Sawin's comment</b>) $n^{B \log^2 n}$ for some explicit constant $B$. So you win! Using Lubotzky's theorem, you still win even if $2$-generator groups are replaced by $d$-generator groups for any fixed $d$.          


Here is Lubotzky's paper: https://core.ac.uk/download/pdf/82373534.pdf