An explicit theory of ${\pi }_1^{\text{un},\text{crys}}({\mathbb{P}}^1-\{0,{\mu }_N,\mathrm{\infty }\})$ - V-1 : The Frobenius extended to ${\pi }_1^{\text{un},\text{DR}}({\mathbb{P}}^1-\{0,{\mu }_{p^{\alpha }N},\mathrm{\infty }\})$

Let $p$ a prime number. For all $N\in {\mathbb{N}}^{\ast }$ prime to $p$, let $k_N$ be a finite field of characteristic $p$ containing a primitive $N$-th root of unity. Let $X_{k_{N},N}=$$$\mathbb{P}$^{1} - ({0,\infty} \cup \mu_{N})$$/$$k_{N}$. This work is an explicit theory of the crystalline pro-unipotent fundamental groupoid $({\pi }_1^{\text{un},\text{crys}})$ of $X_{k_N,N}$. In the parts I to IV, we have considered each possible value of $N$ separately. The purpose of part V is to study the role of the morphisms relating ${\pi }_1^{\text{un}}({\mathbb{P}}^1-\{0,{\mu }_{N_1},\mathrm{\infty }\})$ and ${\pi }_1^{\text{un}}({\mathbb{P}}^1-\{0,{\mu }_{N_2},\mathrm{\infty }\})$ when $N_1$ divides $N_2$. In V-1, we specify this question to the theme of part I, the computation of the Frobenius. For any $N\in {\mathbb{N}}^{\ast }$, let $K_N={\mathbb{Q}}_p({\xi }_N)$ where ${\xi }_N\in \overline{{\mathbb{Q}}_p}$ is a primitive $N$-th root of unity, and $X_{K_{N},N} = $\mathbb{P}$^{1} - ({0,\infty} \cup \mu_{N})$$/$$K_{N}$\mathrm{.}For$N$primeto$p$, we are used to view the Frobenius of ${\pi }_1^{\text{un},\text{crys}}(X_{k_N,N})$ as a structure on ${\pi }_1^{\text{un},\text{DR}}(X_{K_N,N})$. In V-1, we show that the Frobenius of ${\pi }_1^{\text{un},\text{DR}}(X_{K_N,N})$, iterated $\alpha \in $\mathbb{N}$^{\ast}$times,canbeextendedcanonicallyasastructureof$\pi_{1}^{\un,\DR}(X_{K_{p^{\alpha}N},p^{\alpha}N})$. This allows to define generalizations of adjoint $p$-adic multiple zeta values associated with roots of unity of order $p^{\alpha }N$, and several related objects. This also gives a canonical framework to relate to each other the direct method of computation of the Frobenius of I-1 and the indirect methods of computation of the Frobenius of I-2 and I-3.