For connected reductive complex groups G, the smooth locus of the G-character variety of a free group has homotopy groups pi_k(G)^r x pi_{k-1}(PG) in a stable range, and the only CI groups are those whose derived subgroup is a product of special linear groups.
Fundamental groups of character varieties: surfaces and tori
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Bad Representations and Homotopy of Character Varieties
For connected reductive complex groups G, the smooth locus of the G-character variety of a free group has homotopy groups pi_k(G)^r x pi_{k-1}(PG) in a stable range, and the only CI groups are those whose derived subgroup is a product of special linear groups.