Exact triangle and spectral sequence of the equivariant Floer homology of double branched covers

In [OS05] the authors proved an exact triangle for the Heegaard Floer homology of the double branched covers associated to a skein sequence of links, and constructed a spectral sequence from the reduced Khovanov homology to the Heegaard Floer homology of the double branched cover of a link. For $L\subset S^3$ a link and $\mathrm{\Sigma }(L)$ its double branched cover, the invariant $HF(\mathrm{\Sigma }(L))$ doesn’t capture the information of the natural involution on $\mathrm{\Sigma }(L)$. In this paper we study the equivariant version of $HF(\mathrm{\Sigma }(L))$ and prove that similar exact triangle and spectral sequence hold for this equivariant version.