  We consider precompact sequential and Fr\'echet group topologies and show
that some natural constructions of such topologies always result in metrizable
groups answering a question of D.~Dikranjan et al. We show that it is
consistent that all sequential precompact topologies on countable groups are
Fr\'echet (or even metrizable). For some classes of groups (for example
boolean) extra set-theoretic assumptions may be omitted (although in this case
such groups do not have to be metrizable).
  We also build (using $\diamondsuit$) an example of a countably compact
Fr\'echet group that is not $\alpha_3$ and obtain a counterexample to a
conjecture of D.~Shakhmatov as a corollary.
