  Let $M^3$ be a 3-dimensional manifold with fundamental group $\pi_1(M)$ which
contains a quaternion subgroup $Q$ of order 8. In 1979 Cappell and Shaneson
constructed a nontrivial normal map $ f\colon M^3\times T^2\to M^3\times S^2$
which cannot be detected by simply connected surgery obstructions along
submanifolds of codimension 0, 1, or 2, but it can be detected by the
codimension 3 Kervaire-Arf invariant.
  The proof of non-triviality of $\sigma(f)\in L_5(\pi_1(M))$ is based on
consideration of a Browder-Livesay filtration of a manifold $X$ with
$\pi_1(X)\cong \pi_1(M)$. For a Browder-Livesay pair $Y^{n-1}\subset X^n$, the
restriction of a normal map to the submanifold $Y$ is given by a partial
multivalued map $\Gamma\colon L_n(\pi_1(X))\to L_{n-1}(\pi_1(Y))$, and the
Browder-Livesay filtration provides an iteration $\Gamma^n$. This map is a
basic step in the definition of the iterated Browder-Livesay invariants which
give obstructions to realization of surgery obstructions by normal maps of
closed manifolds.
  In the present paper we prove that $\Gamma^3(\sigma(f))=0$ for any
Browder-Livesay filtration of a manifold $X^{4k+1}$ with $\pi_1(X)\cong Q$.
  We compute splitting obstruction groups for various inclusions $\rho\to Q$ of
index 2, describe natural maps in the braids of exact sequences, and make more
precise several results about surgery obstruction groups of the group $Q$.
