The most basic nerve lemma can be found as Corollary 4G.3 in Hatcher's *Algebraic Topology*:

If $\mathcal U$ is an open cover of a paracompact space $X$ such that every nonempty intersection of finitely many sets in $\mathcal U$ is contractible, then $X$ is homotopy equivalent to the nerve $N \mathcal U$.

Many versions of this lemma are summarized in this paper by Heal; note in particular those versions in which the sets in $\mathcal U$ need not be open, such as Borsuk's version, where $\mathcal U$ is a finite regular cover by *closed subsets* of a compact space $X$. The version that I have found with the weakest hypotheses on $\mathcal U$ is in this paper by Nagórko as Theorems 3.3 and 3.4.

The lemma may also be strengthened by providing an *equivariant homotopy equivalence* in the case where $X$ is acted upon by some group $G$ and $\mathcal U$ is invariant under the action. Such a lemma should have the form

If $X$ is acted upon by $G$ such that $\mathcal U$ is a covering by not-necessarily-open subsets, then under some hypotheses on $X$, $G$, and $\mathcal U$, there is a $G$-equivariant homotopy equivalence $X \to N\mathcal U$.

Such lemmas appear as Theorem 4.6 in this paper by González & González, Lemma 2.5 in this paper by Hess & Hirsch, and Proposition 2.2 in this paper by Paris.

The title question of this post may be understood in three ways, all of which I am interested in:

Is there a equivariant version of Nagórko's nerve lemma?

Is there a nerve lemma with hypotheses on $\mathcal U$ weaker than Nagórko's?

What is the equivariant nerve lemma with the weakest hypotheses on $X$, $G$, and $\mathcal U$?