Projective special linear group of degree two is ambivalent iff -1 is a square

Statement
Suppose $$k$$ is a field and $$G = PSL(2,k)$$ is the fact about::projective special linear group of degree two over $$k$$. Then, $$G$$ is an fact about::ambivalent group if and only if $$-1$$ is a square in $$k$$.

In particular, when $$k$$ is a finite field with $$q$$ elements, this is equivalent to saying that $$G$$ is an ambivalent group if and only if $$q$$ is a power of $$2$$ or $$q \equiv 1 \pmod 4$$.

Related facts

 * Special linear group of degree two has a class-inverting automorphism
 * Special linear group of degree two is ambivalent iff -1 is a square
 * Projective special linear group of degree two has a class-inverting automorphism