  A holomorphic torsion invariant of K3 surfaces with involution was introduced
by the second-named author. In this paper, we completely determine its
structure as an automorphic function on the moduli space of such K3 surfaces.
On every component of the moduli space, it is expressed as the product of an
explicit Borcherds lift and a classical Siegel modular form. We also introduce
its twisted version. We prove its modularity and a certain uniqueness of the
modular form corresponding to the twisted holomorphic torsion invariant. This
is used to study an equivariant analogue of Borcherds' conjecture.
