  A two-dimensional chiral conformal field theory can be viewed mathematically
as the representation theory of its chiral algebra, a vertex operator algebra.
Vertex operator algebras are especially well suited for studying logarithmic
conformal field theory (in which correlation functions have logarithmic
singularities arising from non-semisimple modules for the chiral algebra)
because of the logarithmic tensor category theory of Huang, Lepowsky, and
Zhang. In this paper, we study not-necessarily-semisimple or rigid braided
tensor categories $\mathcal{C}$ of modules for the fixed-point vertex operator
subalgebra $V^G$ of a vertex operator (super)algebra $V$ with finite
automorphism group $G$. The main results are that every $V^G$-module in
$\mathcal{C}$ with a unital and associative $V$-action is a direct sum of
$g$-twisted $V$-modules for possibly several $g\in G$, that the category of all
such twisted $V$-modules is a braided $G$-crossed (super)category, and that the
$G$-equivariantization of this braided $G$-crossed (super)category is braided
tensor equivalent to the original category $\mathcal{C}$ of $V^G$-modules. This
generalizes results of Kirillov and M\"{u}ger proved using rigidity and
semisimplicity. We also apply the main results to the orbifold rationality
problem: whether $V^G$ is strongly rational if $V$ is strongly rational. We
show that $V^G$ is indeed strongly rational if $V$ is strongly rational, $G$ is
any finite automorphism group, and $V^G$ is $C_2$-cofinite.
