Given a projective space with standard coordinates over a field. A collineation is (when I understand it correctly) a projective linear transformation of the coordinates, followed by an automorphism of the field. E.g. in the one-dimensional case over $C$ a collineation could be written as either $z \rightarrow \frac{az + b}{cz+d}$ (:= linear projective) or $z \rightarrow \frac{a\bar{z} + b}{c\bar{z}+d}$ (:= linear anti-projective), $a,b,c,d \in C$ where $z$ denotes the non-homogenous coordinate as in $(z,1)$.
Suppose in the one-dimensional case over $C$ a collineation $\phi$ is given by three points and their images: $\phi: ABC \rightarrow A'B'C'$. According to the fundamental theorem $\phi$ is uniquely defined.
Is it now possible to determine whether the automorphism over $C$ is the identity or not, i.e. whether the collineation is projective or anti-projective?
How about the 2- or 3-dimensional case?
A projective transformation in a one-dimensional projective space is uniquely defined by three points and their images. But if the underlying field has a non-trivial automorphism, then every such triple might as well be the result of a projective transformation (possibly identity) combined with that automorphism.
So given three points and their images in $\mathbb{CP}^1$, there are actually two collineations between them, one projective and the other anti-projective.
The same holds for higher dimensions, and other fields, with $d+2$ points (in general position) for $\mathbb{KP}^d$ defining a projective transformation, and all automorphisms of the field $\mathbb K$ providing separate alternatives for collieations.