  In this paper we prove birational superrigidity of finite covers of degree
$d$ of the $M$-dimensional projective space of index 1, where $d\geqslant 5$
and $M\geqslant 10$, with at most quadratic singularities of rank $\geqslant
7$, satisfying certain regularity conditions. Up to now, only cyclic covers
were studied in this respect. The set of varieties with worse singularities or
not satisfying the regularity conditions is of codimension
$\geqslant\frac12(M-4)(M-5)+1$ in the natural parameter space of the family.
