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Let $\Gamma ={\Gamma }_1{*}_G{\Gamma }_2$ be the pushout of two groups ${\Gamma }_i$, i = 1,2, over a common subgroup G, and H be the double mapping cylinder of the corresponding diagram of classifying spaces $B{\Gamma }_1\leftarrow BG\leftarrow B{\Gamma }_2$. Denote by ξ the diagram $I\frac{p}{\leftarrow }H\frac{1}{\to }X=H$, where p is the natural map onto the unit interval. We show that the $Ni{l}^{\sim }$ groups which occur in Waldhausen’s description of $K_{*}\left(\mathbb{Z}\Gamma \right)$ coincide with the continuously controlled groups ${}_{*}^{cc}\left(\xi \right)$, defined by Anderson and Munkholm. This also allows us to identify the continuously controlled groups ${}_{*}^{cc}\left({\xi }^{+}\right)$ which are known to form a homology...