Group embeddings related to automorphism groups of algebraic varieties

The following questions are in the focus of exploration of automorphism groups of algebraic varieties, which is the last decade trend:
(Q1) Is a given group embeddable in the automorphism group of an algebraic variety?
(Q2) If yes, what are the properties of such varieties? Do they exist in some distinguished classes of varieties (e.g., rational, nonrational, affine, complete, etc.)? What are the "extreme"" values of the parameters of such varieties (e.g., the minimum of their dimensions)?
(Q3) Conversely, in which groups can the automorphism groups of algebraic varieties of some type be embedded (e.g., are these groups linear)?
The topics of the talk belong to this blend of abstract algebra and algebraic geometry.