Conjugations in $L^2(\mathcal{H})$

Conjugations commuting with ${\mathbf{M}}_z$ and intertwining ${\mathbf{M}}_z$ and ${\mathbf{M}}_{\bar{z}}$ in $L^2(\mathcal{H})$, where $\mathcal{H}$ is a Hilbert space, are characterized. We also investigate which of them leave invariant the whole Hardy space $H^2(\mathcal{H})$ or a model space $K_{\mathrm{\Theta }}=H^2(\mathcal{H})\ominus \mathrm{\Theta }{H}^2(\mathcal{H})$, where $\mathrm{\Theta }$ is a pure operator valued inner function.