  We study projective fourfolds of $K3^{[2]}$-type with a symplectic involution
and the deformations of their quotients, called orbifolds of Nikulin types;
they are IHS orbifolds. We compute the Riemann--Roch formula for Weil divisors
on such orbifolds and describe the first complete family of orbifolds of
Nikulin type with a polarization of degree $2$ as double covers of special
complete intersections $(3,4)$ in $\mathbb{P}^6$.
