  Let $X_{0}=\mathbb{P}^{1} - (\{0,\infty\} \cup \mu_{N})\text{ }/\text{
}\mathbb{F}_{q}$, with $N \in \mathbb{N}^{\ast}$ and $\mathbb{F}_{q}$ of
characteristic $p>0$ and containing a primitive $N$-th root of unity. We
establish an explicit theory of the crystalline pro-unipotent fundamental
groupoid of $X_{0}$. In part I, we have computed explicitly the Frobenius, and
in particular cyclotomic $p$-adic multiple zeta values. In part II, we use part
I to understand the algebraic relations of cyclotomic $p$-adic multiple zeta
values via explicit formulas ; this is in particular a study of the harmonic
Ihara actions and the maps of comparisons between them introduced in I-2 and
I-3. In II-1, we have developed the basics of algebraic theory of cyclotomic
sequences of prime weighted multiple harmonic sums and adjoint cyclotomic
multiple zeta values viewed as variants of those of the algebraic theory of
cyclotomic multiple zeta values. In this II-2, we use part I and II-1 to show
that one can read some standard algebraic relations of cyclotomic $p$-adic
multiple zeta values via the explicit formulas and via the standard algebraic
relations of sequences of multiple harmonic sums. This amounts to say that the
harmonic Ihara actions and the comparison maps are compatible with algebraic
relations. The two main results are two "harmonic" versions of
Besser-Furusho-Jafari's theorem that $p$-adic multiple zeta values satisfy the
regularized double shuffle relations. This gives two different answers to what
we could call "the adjoint variant" of a question of Deligne and Goncharov
about reading the quasi-shuffle relation of $p$-adic multiple zeta values via
explicit formulas.
