  In this paper, we study $G$-equivariant tensor categories for a finite group
$G$. These categories were introduced by Turaev under the name of $G$-crossed
categories; the motivating example of such a category is the category of
twisted modules over a vertex operator algebra $V$ with a finite group of
automorphisms $G$. We discuss the notion of "orbifold quotient" of such a
category (in the example above, this quotient is the category of modules over
the subalgebra of invariants $V^G$). We introduce an extended Verlinde algebra
for a $G$-equivariant tensor category and give a simple description of the
Verlinde algebra of the orbifold category in terms of the extended Verlinde
algebra of the original category.
  We define an analog of $s,t$ matrices for the extended Verlinde algebra and
show that if $s$ is invertible, then these matrices define an action of
$SL_2(Z)$ on the extended Verlinde algebra. We also show that the $s$-matrix
interchanges tensor product with a much simpler product ("convolution
product"), which can be used to compute the tensor product multiplicities.
