  Consider a group $\Gamma$ acting on a formal (Fedosov) deformation
quantization $\mathbb{A}_\hbar(M)$ of a symplectic manifold $(M,\omega)$. This
canonically induces an action of $\Gamma$ by symplectomorphisms on $M$. We
examine the reverse problem of extending group actions by symplectomorphisms to
the deformation quantization. To do this we first define a notion of extension
that does not impose restrictions on the Fedosov connection realizing
$\mathbb{A}_\hbar(M)$ in its gauge equivalence class by considering composition
of pull-back with certain inner automorphisms of sections of the Weyl bundle.
We reformulate the well known sufficient conditions for existence of such
extensions and show by way of example that they are not necessary. We then turn
to the corresponding classification problem of extensions of a given action by
symplectomorphisms. To this end we associate a group $\bar{\mathcal{G}_\nabla}$
to each Fedosov connection $\nabla$ and show that the (non-Abelian) group
cohomology $\mbox{H}^1(\Gamma,\bar{\mathcal{G}_\nabla})$ classifies extensions
up to a natural notion of equivalence. Finally we develop several tools that
allow for computation of $\mbox{H}^1(\Gamma,\bar{\mathcal{G}_\nabla})$ and
apply these to compute $\mbox{H}^1(\Gamma,\bar{\mathcal{G}_\nabla})$ in a range
of examples.
