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We investigate the $8$-dimensional Riemannian Lie groups $G^8$, carrying aleft-invariant, conformal and minimal foliation $\mathcal{F}$, with leavesdiffeomorphic to the subgroup $\textbf{SU}(2) \times \textbf{SU}(2)$ of $G^8$.Such groups have been classified by E. Ghandour, S. Gudmundsson and T. Turnerin their recent work. They show that these $8$-dimensional Lie groups form areal $13$-dimensional family.  For each left-invariant Hermitian structure $J_\mathcal{V}$ on$\textbf{SU}(2) \times \textbf{SU}(2)$, we extend this to an almost Hermitianstructures $J$ on $G$ adapted to the foliation $\mathcal{F}$ i.e. respectingthe leaf structure on $G$ induced by $\mathcal{F}$. We then classify those$8$-dimensional Lie groups $G$ for which the almost Hermitian structures $J$are integrable ($\mathcal{W}_3\oplus\mathcal{W}_4$), semi-K\"ahler($\mathcal{W}_4$), locally conformal K\"ahler ($\mathcal{W}_3$) or evenK\"ahler ($\mathcal{K}$).  It turns out that for each $J_\mathcal{V}$ we obtain a $9$-dimensional familyof Lie groups $G$ for which $J$ is integrable. In the case of $J$ beingsemi-K\"ahler, we yield a $3$-dimensional family of such groups. We then showthat in the cases of $J$ being K\"ahler or locally conformal K\"ahler there areno solutions i.e. such $8$-dimensional Lie groups do not exist.

Eight-Dimensional Hermitian Lie Groups Conformally Foliated by Minimal $\textbf{SU}(2) \times \textbf{SU}(2)$ Leaves