In this article, we develop tools for computing $G$-crossed extensions of braided tensor categories, including all coherence data (associator, braiding, tensor structures). Their equivariantisations describe representation categories of fixed-point vertex operator algebras, also called orbifolds. The seminal work of Etingof, Nikshych and Ostrik asserts the existence and uniqueness of the $G$-crossed extensions we construct. In particular, for the group $G=\mathbb{Z}_2$ acting as inversion on a discriminant form or metric group, we explicitly construct a braided $\mathbb{Z}_2$-crossed tensor category with a nondegenerate braiding that generalises the Tambara-Yamagami category, now with more than one simple object in the twisted sector. Its equivariantisation yields the modular tensor category of the orbifold of a lattice vertex operator algebra under a lift of $-\mathrm{id}$. In order to derive the above result, we show that $G$-crossed extensions and condensations by commutative algebras commute appropriately, leading to an effective method for constructing new $G$-crossed extensions. Based on this, we also outline a strategy for addressing the general problem of lattice orbifolds.