Sobolev mappings and moduli inequalities on Carnot groups

In the article we study mappings of Carnot groups satisfy moduli inequalities. We prove that homeomorphisms satisfy the moduli inequalities ($Q$-homeomor-phisms) with a locally integrable function $Q$ are Sobolev mappings. On this base in the frameworks of the weak inverse mapping theorem we prove that mappings inverse to Sobolev homeomorphisms of finite distortion of the class $W_{\nu ,\text{loc}}^1(\mathrm{\Omega };{\mathrm{\Omega }}^{\mathrm{\prime }})$ belong to the Sobolev class $W_{1,\text{loc}}^1({\mathrm{\Omega }}^{\mathrm{\prime }};\mathrm{\Omega })$.