The authors deal with the question whether an element of the Wall group \(L_n(\pi_1(X))\) is the surgery obstruction of a degree-one normal map \(f: M\to X\) of closed \(n\)-manifolds \(M\) and \(X\). In the case where the fundamental group \(\pi_1(X)\) is a 2-group, the authors describe new invariants which allow to obtain theorems about elements of the Wall group that cannot be realized by normal maps. Their approach is based on the idea of iterated Browder-Livesay invariants developed by Hambleton and Kharshiladze, as well as on the results about closed manifold surgery problem in oriented case obtained by Hambleton, Milgram, Taylor, and Williams. The authors give applications of their results to the surgery on filtered manifolds, in particular, they show that some elements of the Wall group cannot be realized by normal maps to filtered manifolds.