Let $\mathbb{F}$ be a finite field, $V$ be a vector space over $\mathbb{F}$ and $\sigma$ be a field automorphism of $\mathbb{F}$. I would like to know if the following objects have been considered in the past: I'm looking for a kind of "twisted" quadratic form in the sense that the following holds:
$f \colon V \rightarrow \mathbb{F}$,
$f(x)=x^t A (\sigma(x))$, for $x \in V$, and an arbitrary matrix $A$ of coefficients. Here I denote by $\sigma(x)=(\sigma(x_1),\dots,\sigma(x_n))$, so the automorphism is just applied to every element. Of course for the trivial automorphism, we recover the usual quadratic forms and if the automorphism has order $2$ we get connections to Hermitian forms from what I've seen.
My question is: Has this more general case with an arbitrary automorphism been studied? In particular, are there classification results like for quadratic forms over finite fields, or statements on the number of elements for which $f(x)=a$?