  We study the conjugation involution in Seiberg-Witten theory in the context
of the Ozsv\'ath-Szab\'o and Bloom's spectral sequence for the branched double
cover of a link $L$ in $S^3$. We prove that there exists a spectral sequence of
$\mathbb{F}[Q]/Q^2$-modules (where $Q$ has degree $-1$) which converges to
$\widetilde{\mathit{HMI}}_*(\Sigma(L))$, an involutive version of the monopole
Floer homology of the branched double cover, and whose $E^2$-page is a version
of Bar Natan's characteristic two Khovanov homology of the mirror of $L$. We
conjecture that an analogous result holds in the setting of
$\mathrm{Pin}(2)$-monopole Floer homology.
