Summary: Let \(G\) be a linear algebraic group defined over an infinite field \(k\) (\(\text{char }k\neq 2\)), almost simple and isotropic over \(k\), and \(W(G,k) = G(k)/G(k)^ +\) be the Whitehead group of \(G\). Suppose that \(G\) is not simple connected and \(\pi:\widetilde G\to G\) is a universal covering map (\(\widetilde G\) is simple connected and \(k\)-defined and \(\pi\) is a covering \(k\)-homomorphism). It induces a homomorphism \(\pi_ W: W(\widetilde G,k)\to W(G,k)\). The question arises: is \(\pi_ W\) injective or not? It is proved that \(\pi_ W\) is not injective in general.