dirac matrices - Are Lifshitz and Berestetskii right in this case?

In the Quantum electrodynamics book (look at the problem) its authors Lifshitz and Berestetskei claim that operator of charge conjugation $\hat {C} = -\alpha_{2}$ in Majorana basis transforms as $\hat {C}^{M} = \alpha_{2}$. Here (they started from the standart (Dirac) representation of the gamma-matrices) $$\alpha_{2} = \begin{pmatrix} 0 & \sigma_{y}\\ \sigma_{y} & 0\end{pmatrix}, \quad \beta = \begin{pmatrix} 1 & 0\\ 0 & -1\end{pmatrix},$$ $$\hat {C}^{M} = \hat {U}^{+} \hat {C} \hat {U}, \quad \hat {U} = \frac{1}{\sqrt{2}}(\alpha_{2} + \beta ) = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & \sigma_{y}\\ \sigma_{y} & -1\end{pmatrix} = \hat {U}^{+}.$$ I tried to get their result, but I only got $$\hat {C}^{M} = \frac{1}{2}\begin{pmatrix} 1 & \sigma_{y}\\ \sigma_{y} & -1\end{pmatrix}\begin{pmatrix} 0 & \sigma_{y}\\ \sigma_{y} & 0\end{pmatrix}\begin{pmatrix} 1 & \sigma_{y}\\ \sigma_{y} & -1\end{pmatrix} = \begin{pmatrix} 1 & 0\\ 0 & -1\end{pmatrix},$$ because $\sigma_{y}^{2} = 1$.

Where is the mistake?

An edit.

I found the mistake. I used the wrong definition of transformation of charge conjugation operator under unitary spinor transformation. The correct one is $\hat {C}^{'} = \hat {U}\hat {C}\hat {U}^{T}$.