The Functional Equation $ f \left( x ^ 2 \right) = f ( x ) ^ 2 $

Question

Consider the following functional equation: $$ f \left( x ^ 2 \right) = f ( x ) ^ 2 \text , $$ where $ f : \mathbb Z \to \mathbb Z $.

The only two solutions I could find so far are $ f ( x ) = 1 $ and $ f ( x ) = 0 $.

Are there any other solutions?