Various instances of Harish-Chandra pairs

In this paper we address several algebraic constructions in the context of groupoids, algebroids and $\mathbb{Z}$-graded manifolds. We generalize the results of integration of $\mathbb{N}$-graded Lie algebras to the honest $\mathbb{Z}$-graded case and provide some examples of application of the technique based on Harish-Chandra pairs. We extend the construction to the algebroids setting, the main example being the action Lie algebroid.