  The goal of this article and its precursor is to demonstrate, by example, the
existence of "transfer operators" betweeen relative trace formulas, which
generalize the scalar transfer factors of endoscopy. These transfer operators
have all properties that one could expect from a trace formula comparison:
matching, fundamental lemma for the Hecke algebra, transfer of (relative)
characters. Most importantly, and quite surprisingly, they appear to be of
abelian nature (at least, in the low-rank examples considered in this paper),
even though they encompass functoriality relations of non-abelian harmonic
analysis. Thus, they are amenable to application of the Poisson summation
formula in order to perform the global comparison. Moreover, we show that these
abelian transforms have some structure -- which presently escapes our
understanding in its entirety -- as deformations of well-understood operators
when the spaces under consideration are replaced by their "asymptotic cones".
  In this second paper we use Rankin-Selberg theory to prove the local transfer
behind Rudnick's 1990 thesis (comparing the stable trace formula for
$\operatorname{SL}_2$ with the Kuznetsov formula) and Venkatesh's 2002 thesis
(providing a "beyond endoscopy" proof of functorial transfer from tori to
$\operatorname{GL}_2$). As it turns out, the latter is not completely disjoint
from endoscopic transfer -- in fact, our proof "factors" through endoscopic
transfer. We also study the functional equation of the symmetric-square
$L$-function for $\operatorname{GL}_2$, and show that it is governed by an
explicit "Hankel operator" at the level of the Kuznetsov formula, which is also
of abelian nature. A similar theory for the standard $L$-function was
previously developed (in a different language) by Jacquet.
