The usual $G$-equivariant homology and cohomology groups of a space $X$ with $G$-action are given by the Borel construction: $$H_\ast^G(X)=H_\ast((X\times EG)/G),$$ $$H^\ast_G(X)=H^\ast((X\times EG)/G).$$ Neither of these groups captures the notion of a "homotopically $G$-invariant cycle on $X$". That notion is instead captured by the group $$H_{G,\ast}(X)=H_G^\ast(DX)$$ where $D$ denotes Spanier--Whitehead dual. Do these groups $H_{G,\ast}$ have a standard name and/or notation? They are evidently the place where $G$-equivariant virtual fundamental classes live.
Remark #1: Recall that for any spectrum $X$, there is a canonical isomorphism $H_\ast(X)=H^{-\ast}(DX)$. The point of the question above is that this isomorphism does not extend to equivariant (co)homology, leading to two different reasonable definitions of $G$-equivariant homology and cohomology.
Remark #2: It is unfortunate that the standard notation $H^G_\ast$ refers to a group measuring "$G$-co-invariant cycles", while the group I have denoted $H_{G,\ast}$ above measures "$G$-invariant cycles".
