  It has been shown that the proper, non-locally finite pseudovarieties of
abelian groups are not tame with respect to the canonical signature. In this
paper, we show that every decidable, proper, non-locally finite pseudovariety
of abelian groups is completely tame with respect to a further enlarged
implicit signature. This theorem yields as a corollary that a pseudovariety of
abelian groups is decidable if and only if it is completely tame.
