  Let $P$ be a Poisson algebra, $E$ a vector space and $\pi : E \to P$ an
epimorphism of vector spaces with $V = {\rm Ker} (\pi)$. The global extension
problem asks for the classification of all Poisson algebra structures that can
be defined on $E$ such that $\pi : E \to P$ becomes a morphism of Poisson
algebras. From a geometrical point of view it means to decompose this groupoid
into connected components and to indicate a point in each such component. All
such Poisson algebra structures on $E$ are classified by an explicitly
constructed classifying set ${\mathcal G} {\mathcal P} {\mathcal H}^{2} \, (P,
\, V)$ which is the coproduct of all non-abelian cohomological objects
${\mathcal P} {\mathcal H}^{2} \, (P, \, (V, \cdot_V, [-,-]_V))$ which are the
classifying sets for all extensions of $P$ by $(V, \cdot_V, [-,-]_V)$. The
second classical Poisson cohomology group $H^2 (P, V)$ appears as the most
elementary piece among all components of ${\mathcal G} {\mathcal P} {\mathcal
H}^{2} \, (P, \, V)$. Several examples are provided in the case of metabelian
Poisson algebras or co-flag Poisson algebras over $P$: the latter being Poisson
algebras $Q$ which admit a finite chain of epimorphisms of Poisson algebras
$P_n : = Q \stackrel{\pi_{n}}{\longrightarrow} P_{n-1} \, \cdots \, P_1
\stackrel{\pi_{1}} {\longrightarrow} P_{0} := P$ such that ${\rm dim} ( {\rm
Ker} (\pi_{i}) ) = 1$, for all $i = 1, \cdots, n$.
