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

For example, https://arxiv.org/pdf/1811.06190.pdf provides an algorithm computing the generator of a centralizer. In $n=2$ people have studied them extensively, there are a lot of references.

But not what I am interested in the following: I want to find a group of following or at least find its property: $C(K)=\{ B \in GL(n, Z) |B^{T}KB=K \}.$ 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 element$B$ will determine a quadratic surface and we are asking for a set of solutions which are the 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 $GL(n, Z)$ is approximate $e^{nlogn^{1/2}}$ http://math.univ-lyon1.fr/~nicolas/JofAlgebra98.pdf Other conjugacy property 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