  This paper is the second part of the series "Spherical higher order Fourier
analysis over finite fields", aiming to develop the higher order Fourier
analysis method along spheres over finite fields, and to solve the geometric
Ramsey conjecture in the finite field setting.
  In this paper, we study additive combinatorial properties for shifted
modules, i.e. the structure of sets of the form $E\hat{+} E$, where $E$ is a
collection of shifted modules of the polynomial ring
$\mathbb{R}[x_{1},\dots,x_{d}]$ and we identify two modules if their difference
contains the zero polynomial. We show that under appropriate definitions, the
set $E\hat{+} E$ enjoys properties similar to the conventional setting where
$E$ is a subset of an abelian group. In particular, among other results, we
prove the Balog-Gowers-Szemer\'edi theorem, the Rusza's quasi triangle
inequality and a weak form of the Pl\"unnecke-Rusza theorem in the setting of
shifted modules. We also show that for a special class of maps $\xi$ from
$\mathbb{Z}_{K}^{d}$ to the collection of all shifted modules of
$\mathbb{R}[x_{1},\dots,x_{d}]$, if the set
$\xi(\mathbb{Z}_{K}^{d})+\xi(\mathbb{Z}_{K}^{d})$ has large additive energy,
then $\xi$ is an almost linear Freiman homomorphism. This result is the crucial
additive combinatorial input we need to prove the spherical Gowers inverse
theorem in later parts of the series.
