$K_2$ for the simplest integral group rings and topological applications

We calculate the group $K_2(\Lambda)$, where $\Lambda=\mathbb Z/2[\pi]$ is the group ring of a fundamental group with coefficients in the field $\mathbb Z/2$ and $\pi=\mathbb Z/2\oplus\mathbb Z/2$ is the simplest elementary Abelian group of rank $2$. Using these calculations we estimate from below the value $K_2(\overline\Lambda)$, where $\overline\Lambda$ is the integral group ring of the group $\pi$. This calculation yields certain corollaries in the theory of pseudo-isotopies, since the group $Wh_2(\mathbb Z/2^2)$ turns out to be non-trivial. Constructions in differential topology are discussed that lead to calculations of $Wh_2$-valued invariants.