Similarities of orthogonal and skew orthogonal

Question

Does $A^TA=-I$ has the same number of solutions of 2 by 2 matrices $A$ with $A^TA=I$ in $Z_p$ for a fixed $p>2$ ?

Answer 1

For example: with $p = 5$, we will have $$ A^TA = I \iff (2A)^T(2A) = -I $$ so that the map $A \mapsto 2A$ is a bijection between the solution sets.

A similar trick works whenever $-1$ is a quadratic residue modulo $p$.