## Translating a geometric deformation problem into a deformation functor

I have seen deformation problems given by commutative squares

$\begin{array}.X & \rightarrow & \tilde{X}\\ \downarrow & & \downarrow\\ \mathrm{Spec}\, A & \rightarrow & \mathrm{Spec}\, B\\ \end{array}$

where $X$ is a scheme and $A$ and $B$ are local Artin $\mathbf{k}$-algebras.

My question is, how does one translate a geometric deformation problem, say deformations of a uniform lattice (discrete Lie group) $\Gamma < G$ acting properly discontinuously (cocompactly, whatever) on a homogeneous space $G/H$, into a deformation diagram/functor that resembles the above diagram?

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