Verifying unitary congruence of coninvolutions, skew-coninvolutions, and connilpotent matrices of index two

It is shown that $n\times n$ solutions $A$ and $B$ of the matrix equation
$$ X\overline X=\delta I, $$
where $\delta$ is one and the same scalar for both matrices, are unitarily congruent if and only if
$$ \operatorname{tr}(A^*A)^k=\operatorname{tr}(B^*B)^k,\qquad k=1,2,\dots,n. $$
Bibl. – 8 titles.