Almost all finite groups are $2$-groups of class two

We prove that almost all finite groups are $2$-groups of nilpotency class two, when groups of order at most $x$ are counted up to isomorphism and $x\to\infty$. More precisely, uniformly for $2^m\leq x<2^{m+1}$, all but an $O(2^{-cm^2})$ proportion have order $2^m$ and class two, for some absolute co