An example concerning equivariant deformations

We give an example of $Z_2$-space $X$ with a property that the identity map $\text{\rm{id}}_X:X\to X$ as well as its restriction to the fixed pointset of the group action $\text{\rm{id}}^{Z_2}:X^{Z_2}\to X^{Z_2}$ are deformable to fixed point free maps whereas there is no fixed point freemap in the equivariant homotopy class of the identity $[\text{\rm{id}}_X]_{ Z_2}$.