TITLE: Finite *covering* groups that act freely on some sphere
QUESTION [6 upvotes]: A remarkable result (reviewed here) -- going back, at least, to P. A. Smith, developed by Cartan & Eilenberg and Milnor, and culminating in the theorem of Madsen, Thomas & Wall -- characterizes those finite groups which admit a free continuous action on a topological sphere in at least one dimension.
I am wondering about the following more general question:

Which finite groups are covered by (i.e.: receive a surjective homomorphism from) a finite group that acts freely on some topological sphere?

For example (here), there are dihedral groups of order $2 p$ (for a prime number $p$) which thus violate Milnor's "$2 p$-condition" and hence cannot have any free action on any sphere; but their double cover by the corresponding binary dihedral group does have a free action on the 3-sphere (namely by regarding the binary dihedral group as a subgroup of $\mathrm{SU}(2) \simeq S^3$).
So I am wondering how generic this situation is. Maybe all finite groups can be covered by ones that act freely on some sphere?
Looking at the above example, one sees that the double cover of $\mathrm{SO}(3)$ by $\mathrm{SU}(2)$ remains non-trivial on all non-cyclic subgroups, whereby the order of all non-cyclic subgroups is multiplied by 2 under the extension, so that Smith/Cartan&Eilenberg's $p^2$-condition and Milnor's $2 p$-condition are enforced.
This seems to suggest a strategy for how to understand finite covering groups freely acting on some sphere in more generality. Maybe it's easy, but I am not sure yet.

REPLY [10 votes]: For a group to act freely on a sphere (even on a homotopy sphere), it must not have subgroups of the form $\mathbb Z/p \times \mathbb Z/p$ (this is the $p^2$-condition you mention). It follows that its Sylow subgroups are all cyclic or generalized quaternion.
Thus, for groups whose cover acts freely on a (homotopy) sphere, their Sylow subgroups must be cyclic, generalized quaternion, or dihedral (as a Sylow subgroup of a quotient is a quotient of a Sylow subgroup by Sylow's thorems).
Groups satisfying this condition were called $\mathcal P'$-groups by Wall, and classified in his article On the structure of finite groups with periodic cohomology.
This is only a necessary condition and not (necessarily) sufficient, but it seems very stringent. One could look through this list for examples where the 2p condition fails, to see if it fails also on every cover.