  We obtain additional Diophantine applications of the methods surrounding
Darmon's program for the generalized Fermat equation developed in the first
part of this series of papers. As a first application, we use a multi-Frey
approach combining two Frey elliptic curves over totally real fields, a Frey
hyperelliptic curve over $\mathbb{Q}$ due to Kraus, and ideas from the Darmon
program to give a complete resolution of the generalized Fermat equation $$x^7
+ y^7 = 3 z^n$$ for all integers $n \ge 2$. Moreover, we explain how the use of
higher dimensional Frey abelian varieties allows a more efficient proof of this
result due to additional structures that they afford, compared to using only
Frey elliptic curves.
  As a second application, we use some of these additional structures that Frey
abelian varieties possess to show that a full resolution of the generalized
Fermat equation $x^7 + y^7 = z^n$ depends only on the Cartan case of Darmon's
big image conjecture. In the process, we solve the previous equation for
solutions $(a,b,c)$ such that $a$ and $b$ satisfy certain $2$ or $7$-adic
conditions and all $n \ge 2$.
