A $C_2$ axis of symmetry means that I should be able to rotate the molecule about an axis by $180^\circ$. But I would need a composition of the following two rotations
- $90^\circ$ rotation about $\ce{C=C=C}$ axis
- $180^\circ$ about the perpendicular bisector of $\ce{C=C=C}$.
Would that count as a $C_2$ axis of rotation?