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

An explicit theory of $\pi_{1}^{\un,\crys}(\mathbb{P}^{1} -  \{0,\mu_{N},\infty\})$ - II-2 : From standard algebraic relations of weighted  multiple harmonic sums to those of cyclotomic $p$-adic multiple zeta values