  Consider the abelian category ${\mathcal C}$ of commutative group schemes of
finite type over a field $k$, its full subcategory ${\mathcal F}$ of finite
group schemes, and the associated pro category ${\rm Pro}({\mathcal C})$ (resp.
${\rm Pro}({\mathcal F})$) of pro-algebraic (resp. profinite) group schemes.
When $k$ is perfect, we show that the profinite fundamental group $\varpi_1 :
{\rm Pro}({\mathcal C}) \to {\rm Pro}({\mathcal F})$ is left exact and commutes
with base change under algebraic field extensions; as a consequence, the higher
profinite homotopy functors $\varpi_i$ vanish for $i \geq 2$. Along the way, we
describe the indecomposable projective objects of ${\rm Pro}({\mathcal C})$
over an arbitrary field $k$.
