<p>This paper has two aims: the first is to define a projective Fraïssé family whose limit approximates the universal Knaster continuum. The family is such that the group Aut(<InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({\mathbb K}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mrow> <mi mathvariant="double-struck">K</mi> </mrow> </mrow> </math></EquationSource> </InlineEquation>) of automorphisms of the Fraïssé limit is a dense subgroup of the group, Homeo(<i>K</i>), of homeomorphisms of the universal Knaster continuum.</p><p>The second aim is to compute the universal minimal flows of Aut(<InlineEquation ID="IEq2"> <EquationSource Format="TEX">\({\mathbb K}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mrow> <mi mathvariant="double-struck">K</mi> </mrow> </mrow> </math></EquationSource> </InlineEquation>) and Homeo(<i>K</i>). We prove that both have universal minimal flow homeomorphic to the universal minimal flow of the free abelian group on countably many generators. The computation involves proving that both groups contain an open, normal subgroup which is extremely amenable.</p>