  In this paper we introduce the crossed product construction for a discrete
group action on an operator system. In analogy to the work of E. Katsoulis and
C. Ramsey, we describe three canonical crossed products arising from such a
dynamical system. We describe how these crossed product constructions behave
under $G$-equivariant maps, tensor products, and the canonical $C^*$-covers. We
show that hyperrigidity is preserved under two of the three crossed products.
Finally, using A. Kavruk's notion of an operator system that detects
$C^*$-nuclearity, we give a negative answer to a question on operator algebra
crossed products posed by Katsoulis and Ramsey.
