While it is well known that the moduli space of $G$-bundles over a smooth projective curve is compact, it is not the case for an arbitrary base variety. This motivated the definition of $G$-sheaves by Gomez and Sols who proved that their moduli space is a compactification of the moduli space of $G$-bundles.
In this talk I will study the deformation and obstruction theory of these objects when $G$ is either the symplectic or the orthogonal group. This is joint work with T. Gomez.