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Let $P$ be a Poisson algebra, $E$ a vector space and $\pi : E \to P$ anepimorphism of vector spaces with $V = {\rm Ker} (\pi)$. The global extensionproblem asks for the classification of all Poisson algebra structures that canbe defined on $E$ such that $\pi : E \to P$ becomes a morphism of Poissonalgebras. From a geometrical point of view it means to decompose this groupoidinto connected components and to indicate a point in each such component. Allsuch Poisson algebra structures on $E$ are classified by an explicitlyconstructed 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 theclassifying sets for all extensions of $P$ by $(V, \cdot_V, [-,-]_V)$. Thesecond classical Poisson cohomology group $H^2 (P, V)$ appears as the mostelementary piece among all components of ${\mathcal G} {\mathcal P} {\mathcalH}^{2} \, (P, \, V)$. Several examples are provided in the case of metabelianPoisson algebras or co-flag Poisson algebras over $P$: the latter being Poissonalgebras $Q$ 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 ${\rm dim} ( {\rmKer} (\pi_{i}) ) = 1$, for all $i = 1, \cdots, n$.

The global extension problem, crossed products and co-flag non-commutative Poisson algebras