Reflexivity in precompact groups and extensions

We establish some general principles and find some counter-examples concerning the Pontryagin reflexivity of precompact groups and P-groups. We prove in particular that:(1)A precompact Abelian group G   of bounded order is reflexive iff the dual group G∧G∧ has no infinite compact subsets and every compact subset of G is contained in a compact subgroup of G.(2)Any extension of a reflexive P-group by another reflexive P-group is again reflexive. We show on the other hand that an extension of a compact group by a reflexive ω-bounded group (even dual to a reflexive P-group) can fail to be reflexive.We also show that the P-modification of a reflexive σ-compact group can be non-reflexive (even if, as proved in [20], the P-modification of a locally compact Abelian group is always reflexive).