Using general sieve-type methods of number theory and certain density estimates for prime numbers, the authors derive asymptotic formulae for \(A(n)-C(n)\) and \(N(n)-A(n)\), where \(A(n)=\#\{m\leq n:\) every group of order \(m\) is abelian\(\},\) \(C(n)=\#\{m\leq n:\) every group of order \(m\) is cyclic\(\}\), and \(N(n)=\#\{m\leq n:\) every group of order \(m\) is nilpotent\(\}\). The second author [Arch. Math. 31, 536--538 (1978; Zbl 0388.20021)] and \textit{E. J. Scourfield} [Acta Arith. 29, 401--423 (1976; Zbl 0286.10023)] showed previously that asymptotically all three of the above counting functions have the form \[ (1+o(1))ne^{-\gamma}/\log_3n. \] The present authors now prove that there exist constants \(c_1\), \(c_2\) such that \[ A(n)-C(n)=(1+o(1))c_1n/(\log_2n)(\log_3n)^2, \] \[ N(n)-A(n)=(1+o(1))c_2n/(\log_2n)^2(\log_3n)^2. \]