  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 $W^1_{\nu,\text{loc}}(\Omega;\Omega')$ belong to the Sobolev class
$W^1_{1,\text{loc}}(\Omega';\Omega)$.
