  We use involutive Heegaard Floer homology to extend the Ozsv\'ath-Szab\'o
branched double cover spectral sequence relating a version of Khovanov homology
and the Heegaard Floer homology of branched double covers. Our main tools are
Lipshitz, Ozsv\'ath, and Thurston's reconstruction of the Ozsv\'ath-Szab\'o
spectral sequence using bordered Floer homology and Hendricks and Lipshitz's
surgery exact triangle in involutive bordered Floer homology.
