  Fix a scheme $S$ of characteristic $p$. Let $\mathscr{M}$ be an $S$-algebraic
stack and let $\mbox{Fdiv}(\mathscr{M})$ be the stack of $\mbox{F}$-divided
objects, that is sequences of objects $x_i\in\mathscr{M}$ with isomorphisms
$\sigma_i:x_i\to \mbox{F}^*x_{i+1}$. Let $\mathscr{X}$ be a flat, finitely
presented $S$-algebraic stack and $\mathscr{X}\to \Pi_1(\mathscr{X}/S)$ the
\'etale fundamental pro-groupoid, constructed in the present text. We prove
that if $\mathscr{M}$ is a quasi-separated Deligne-Mumford stack and
$\mathscr{X}\to S$ has geometrically reduced fibres, there is a bifunctorial
isomorphism of stacks \[\mathscr{H}\!om(\Pi_1(\mathscr{X}/S),\mathscr{M})
\simeq \mathscr{H}\!om(\mathscr{X},\mbox{Fdiv}(\mathscr{M})).\] In particular,
the system of relative Frobenius morphisms $\mathscr{X}\to \mathscr{X}^{p/S}\to
\mathscr{X}^{p^2/S}\to\dots$ allows to recover the space of connected
components $\pi_0(\mathscr{X}/S)$ and the relative \'etale fundamental gerbe.
In order to obtain these results, we study the existence and properties of
relative perfection for algebras in characteristic $p$.
