<p>In this article, we study Sobolev homeomorphisms and composition operators on homogeneous Lie groups. We prove that a measurable homeomorphism <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\varphi \colon \Omega \to\widetilde{\Omega}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>φ</mi> <mo lspace="0pt">:</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">→</mo> <mover accent="true"> <mi mathvariant="normal">Ω</mi> <mo stretchy="true">~</mo> </mover> </mrow> </math></EquationSource> </InlineEquation> belongs to the Sobolev space <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(L^{1}_{q}(\Omega; \widetilde{\Omega})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>L</mi> <mi>q</mi> <mn>1</mn> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>;</mo> <mover accent="true"> <mi mathvariant="normal">Ω</mi> <mo stretchy="true">~</mo> </mover> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(1\leq q &lt; \infty\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>q</mi> <mo>&lt;</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, if and only if <InlineEquation ID="IEq14"> <EquationSource Format="TEX">\(\varphi\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>φ</mi> </math></EquationSource> </InlineEquation> generates a bounded composition operator on Sobolev spaces.</p>