  Recall that a space X is selectively pseudocompact if for every sequence
(U_n) of non-empty open subsets of X one can choose a point x_n in U_n for all
n such that the resulting sequence (x_n) has an accumulation point in X. This
notion was introduced under the name strong pseudocompactness by
Garc\'ia-Ferreira and Ortiz-Castillo, the present name is due to
Dorantes-Aldama and the first author. In 2015, Garc\'ia-Ferreira and Tomita
constructed a pseudocompact Boolean group that is not selectively
pseudocompact. We prove that if the subgroup topology on every countable
subgroup H of an infinite Boolean topological group G is finer than its maximal
precompact topology (the so-called Bohr topology of H), then G is not
selectively pseudocompact, and from this result we deduce that many known
examples in the literature of pseudocompact Boolean groups automatically fail
to be selectively pseudocompact. We also show that, under the Singular Cardinal
Hypothesis, every infinite pseudocompact Boolean group admits a pseudocompact
reflexive group topology which is not selectively pseudocompact.

General Topology