  Wiltshire-Gordon has introduced a homotopy model for ordered configuration
spaces on a given simplicial complex. That author asserts that, after a
suitable subdivision, his model also works for unordered configuration spaces.
We supply details justifying Wiltshire-Gordon's assertion and, more
importantly, uncover the equivariant properties of his more-general
simplicial-difference model for the complement of a subcomplex inside a larger
complex. This is achieved by proving an equivariant version of the Nerve Lemma.
In addition, in the case of configuration spaces, we show that a slight
variation of the model has better properties: it is regular and sits inside the
configuration space as a strong and equivariant deformation retract. Our
variant for the configuration-space model comes from a comparison, in the
equivariant setting, between Wiltshire's simplicial difference and a well known
model for the complement of a full subcomplex on a simplicial complex.
