Let Fⁿ be a connected, smooth and closed n-dimensional manifold satisfying the following property: if $N^m$ is any smooth and closed m-dimensional manifold with m > n and $T:N^m\to N^m$ is a smooth involution whose fixed point set is Fⁿ, then m = 2n. We describe the equivariant cobordism classification of smooth actions $(M^m;\Phi )$ of the group $G=Z{₂}^k$ on closed smooth m-dimensional manifolds $M^m$ for which the fixed point set of the action is a submanifold Fⁿ with the above property. This generalizes a result of F. L. Capobianco,...