  Recent LHC bounds on squark masses combined with naturalness and flavor
considerations motivate non-trivial sfermion mass spectra in the supersymmetric
Standard Model. These can arise if supersymmetry breaking is communicated to
the visible sector via new extended gauge symmetries. Such extended symmetries
must be spontaneously broken, or confined, complicating the calculation of soft
masses. We develop a new formalism for calculating perturbative gauge-mediated
two-loop soft masses for gauge groups with arbitrary patterns of spontaneous
symmetry breaking, simplifying the framework of "Higgsed gauge mediation." The
resulting expressions can be applied to Abelian and non-Abelian gauge groups,
opening new avenues for supersymmetric model building. We present a number of
examples using our method, ranging from grand unified threshold corrections in
standard gauge mediation to soft masses in gauge extensions of the Higgs sector
that can raise the Higgs mass through non-decoupling D-terms. We also outline a
new mediation mechanism called "flavor mediation", where supersymmetry breaking
is communicated via a gauged subgroup of Standard Model flavor symmetries.
Flavor mediation can automatically generate suppressed masses for
third-generation squarks and implies a nearly exact U(2) symmetry in the first
two generations, yielding a "natural SUSY" spectrum without imposing ad hoc
global symmetries or giving preferential treatment to particular generations.
