This work studies the problem of equivariant cobordism classification of cohomologically trivial smooth involutions on the \(2k\)-dimensional quaternionic projective space \(HP(2k)\). The authors use the space \(R^m(T)= S^m\times HP(2k)/A\times T\) where \(A\) is the antipodal map on \(S^m\) and \(T\) is a cohomological smooth involution on \(HP(2k)\). There is a 4-dimensional class \(c\in H^4(R^m(T);Z_2)\) where they show that \(S_q^1(c)=0\). The results of the work are obtained using the hypothesis that \(S_q^2(c)=0\). With this hypothesis and the use of characteristic classes the authors get a quite simple classification of the equivariant cobordism of cohomologically trivial smooth involutions and also a description of the fixed point set of the involution. The paper contains few misprints which sometimes can confuse the reader.