  In math.GR/0510298, we showed that every loop isotopic to an F-quasigroup is
a Moufang loop. Here we characterize, via two simple identities, the class of
F-quasigroups which are isotopic to groups. We call these quasigroups
FG-quasigroups. We show that FG-quasigroups are linear over groups. We then use
this fact to describe their structure. This gives us, for instance, a complete
description of the simple FG-quasigroups. Finally, we show an equivalence of
equational classes between pointed FG-quasigroups and central generalized
modules over a particular ring.
