$\DeclareMathOperator\GL{GL}$Let $n$ be a positive integer $\geq 2$. The setting is that $K \in \GL(n,\mathbb{Z})$, and people are interested in understanding the centralizer: $$ C(K)=\{ B \in \GL(n,\mathbb{Z}) \mathrel|B^{-1}KB=K \}. $$

For example, Eick, Hofmann, and O'Brien - The conjugacy problem in $\GL(n, Z)$ provides an algorithm computing the generator of a centralizer. In $n=2$ people have studied them extensively, there are a lot of references.

But I am interested in understanding the variant $$C'(K)=\{ B \in \GL(n,\mathbb{Z}) \mathrel|B^{T}KB=K \}.$$ I just change inverse to be the transpose. Obviously, $C'(K)$ is also a group. Moreover, the $\det(B)=\pm 1$ since one can take determinant on both sides. So I think this problem is much easier than the original problem. Although it seems like all the elements $B$ will determine a quadratic surface and we are asking for a set of solutions which is an arithmetic problem. The following two questions are interesting to me.

1. One obvious question is that, given $K$, is $C'(K)$ a finite group?
2. If so, can one find the maximal order of the group in terms of $n$? For example, in the original conjugate problem, people have shown that the maximal torsion order in $\GL(n,\mathbb{Z})$ is approximately $e^{n\log(n)^{1/2}}$ (Levitt - On the maximum order of torsion elements in $\GL(n, \mathbf Z)$ and $\operatorname{Aut}(F_n)$). A related MO question: Maximal order of finite subgroups of $GL(n,Z)$.

Basically, I just want to ask, did people study this type of problem before instead of the conjugacy? Any partial results or keywords are welcome.