  Biquandles are algebraic objects with two binary operations whose axioms
encode the generalized Reidemeister moves for virtual knots and links. These
objects also provide set-theoretic solutions of the well-known Yang-Baxter
equation. The first half of this paper proposes some natural constructions of
biquandles from groups and from their simpler counterparts, namely, quandles.
We completely determine all words in the free group on two generators that give
rise to (bi)quandle structures on all groups. We give some novel constructions
of biquandles on unions and products of quandles, including what we refer as
the holomorph biquandle of a quandle. These constructions give a wealth of
solutions of the Yang-Baxter equation. We also show that for nice quandle
coverings a biquandle structure on the base can be lifted to a biquandle
structure on the covering. In the second half of the paper, we determine
automorphism groups of these biquandles in terms of associated quandles showing
elegant relationships between the symmetries of the underlying structures.
