  In this paper, which is a continuation of earlier work by the first author
and Gunnar Carlsson, one of the first results we establish is the additivity of
the motivic Becker-Gottlieb transfer, as well as their \'etale realizations.
This extends the additivity results the authors already established for the
corresponding traces. We then apply this to derive several important
consequences: for example, in addition to obtaining the analogues of various
double coset formulae known in the classical setting of algebraic topology, we
also obtain applications to Brauer groups of homogeneous spaces associated to
reductive groups over separably closed fields. We also consider the
relationship between the transfer on schemes provided with a compatible action
by a $1$-parameter subgroup and the transfer associated to the fixed point
scheme of the $1$-parameter subgroup.
