Adjacency preservers on invertible hermitian matrices

Consider the graph with the vertex set formed by all $n\times n$ invertible hermitian matrices with coefficients in a finite field $\mathbb{F}_{q^2}$, where two matrices form an edge $\{A,B\}$ if and only if the rank of $A-B$ is one.

In a seminar talk about a year ago I showed that this graph is a core (the case $q=3$ was not considered), which means that all its endomorphisms are automorphisms.

In this talk I will describe the characterization of all endomorphisms (i.e. automorphisms) for $q\geq 4$.