Any $H_2$-reducible (complex) Hadamard matrix is equivalent to a member of a three-parameter family of dephased matrices $H$ found by B. Karlsson [106] $$K_6^{(3)}=\left[\begin{array}{rrr} F_2 & Z_1 & Z_2\\ Z_3 & \frac{1}{2}Z_3 A Z_1 & \frac{1}{2} Z_3 B Z_2\\ Z_4 & \frac{1}{2}Z_4 B Z_1 & \frac{1}{2} Z_4 A Z_2 \end{array}\right]:\quad A=\left[\begin{array}{rr} A_{11} & A_{12}\\ \bar{A}_{12} & -\bar{A}_{11} \end{array}\right],$$ where $$ A_{11} = -\frac{1}{2}+i\frac{\sqrt{3}}{2}\left(\cos\theta + e^{-i\varphi}\sin\theta\right),\qquad A_{12} = -\frac{1}{2}+i\frac{\sqrt{3}}{2}\left(-\cos\theta + e^{i\varphi}\sin\theta\right),\\ $$ for any $\theta,\varphi\in[0,\pi)$ and $B=-F_2-A$. In the matrices $$Z_k=\left[\begin{array}{rr}1&1\\z_k&-z_k\end{array}\right]$$ for $k\in\{1,2\}$ and their transposed forms $(k\in\{3,4\})$, the elements $z_k$ are related through the Möbius transformation: $$\begin{aligned} z_3^2&=\mathcal{M}_A\left(z_1^2\right),\qquad z_3^2=\mathcal{M}_B\left(z_2^2\right),\\ z_4^2&=\mathcal{M}_A\left(z_2^2\right),\qquad z_4^2=\mathcal{M}_B\left(z_1^2\right), \end{aligned}$$ where $$\mathcal{M}(z)=\frac{\alpha z-\beta}{\beta^*z-\alpha^*}$$ and parameters $\alpha$ and $\beta$ (and their complex conjugates denoted by $^*$) are functions of $A_{kj}$, see [106].
In general, one of the parameters $z_k$ can be chosen freely, say $z_1=\exp(i \psi)$ for $\psi\in[0,\pi)$. Then $$\begin{aligned} z_2^2 &= \mathcal{M}_A^{-1}\left(\mathcal{M}_B\left(z_1^2\right)\right)= \mathcal{M}_B^{-1}\left(\mathcal{M}_A\left(z_1^2\right)\right),\\ z_3^2 &= \mathcal{M}_A^{-1}\left(z_1^2\right),\\ z_4^2 &= \mathcal{M}_B^{-1}\left(z_1^2\right). \end{aligned}$$ Any sign combinations for $z_k$ lead to three-parameter families that are equivalent to each other.