Determine the unrestricted correction term and SR-group counting sequence

Determine an explicit formula for the correction term e(m), which counts SR-groups of order 4m whose normal Hall subgroup of odd order is nonabelian, or determine the unrestricted sequence f(2^k) counting SR-groups of order 2^k.

Background

For an odd integer m, the paper writes f(4m)=h(m)+e(m), where h(m) counts the groups whose odd Hall subgroup is abelian and e(m) counts the remaining SR-groups with nonabelian odd Hall subgroup. The classification determines e(m) throughout the finite range considered and constructs infinite families showing that the correction term has infinite support.

The paper also provides exact reconstruction identities and bounds for the sequence f(2^k), but these do not constitute a closed formula. The authors explicitly leave both the general correction term and the power-of-two counting sequence unresolved.